Lecture Notes for Math 1000 - Memorial University of ...xiangshengw/math1000/math1000.pdf · Explained the material clearly. ... Xiangsheng Wang methods are much better from going

Lecture Notes for Math 1000

Xiang-Sheng Wang

Memorial University of Newfoundland

Office HH-2016 Phone 864-4321

Office hour 1000-1130 Monday Tuesday and Wednesday

Email xswangmunca or xswang4gmailcom

Course website httpwwwucsmunca~xiangshengwmath1000html

Lecture Notes for Math 1000 First Previous Next Last 1

Self-introduction

bull Name Xiang-Sheng Wang

bull Position Teaching postdoctoral fellow

bull PhD received 2009

bull Previous affiliation York University

bull Teaching experience

1 Math 1013 (Applied Calculus I)2 Math 1190 (Introduction to Sets and Logic)3 Math 1540 (Introductory Mathematics for Economists II)4 Math 6378 (Applied Delay Differential Equations graduate course)


What did my previous students say about me

bull Professor presented the material in a way that made it feel was less overwellingExplained the material clearly I thought the professor made a great effort tohave students understand what was being taught in lecture

bull The thing the made me want to come to class is number of examples beingused to demonstrate a theorem and not just doing an easy example and thatbeing the end of it Xiangsheng Wang methods are much better from going toa easy question to a tougher question so we know what we can expect in thehomework Also there is no real rush everything is done properly and everyminute is used well

bull I valued the professorrsquos teaching style He was able to convey concepts andrules very clearly so a student would be able to follow step-by-step in thecalculations and problem solutions



bull Teacher showed me that math is not impossible

bull The Professor took EXTRA SPECIAL interest in teaching us He didnt justrandomly scroll through presentation slides He put up MANY examplessumson the chalk board which we could write down in our notebooks amp revise laterNO OTHER PROFESSOR has done that till date Especially in a subject likeMath this is of prime importance He made sure each one of us understoodwhat he was teaching

bull Professor Wang is able to deliever the course material effectively and efficientlyHe provides many examples to ensure we understand the material Amazingprof



bull I found it easier to understand everything from Professor Xiangsheng Wang andI think overall everyone is doing better in this class Also the instructor goesout his way to answer questions through email which is very convenient whenyou can not make the office hours

bull The Professor loves doing what he does best - teaching He would allot time tous students even outside his office hours to explain us the material

bull The Professor set exam papers which checked our knowledge about thematerial not unnecessarily tough questions to pressure us

bull The Professor not only wrote many examples on the board along withdefinitions etc but also presented slides and explained the concepts therein indetail once again which were concise amp useful for later reference


What can you benefit from me

bull I will show you that mathematics is not impossible I will try my best efforts tohelp you understand the materials in this course by using many illustrativeexamples

bull I am always willing to answer your questions So please do not hesitate tocome to me whenever you have any

bull My contact information

Office HH-2016Office hour MTW 1000-1130 (or by appointment)Phone 864-4321Email xswang4gmailcom


Outline

bull Textbook (either of the following two)

1 Calculus Early Transcendentals with WebAssign Access Card (bundle) byJon Rogawski OR

2 Math 1000 1001 Course Notes by Bruce Watson +Calculus Early Transcendentals WebAssign Card by Jon Rogawski

bull Topics to be covered limits and continuity differentiation and applications ofdifferentiation

bull Prerequisite Math 1090 or a combination of placement test and high schoolMathematics scores acceptable to the department

bull Marks 55 Final Exam 30 In-class tests (three tests with 10 each) 75WebAssign 75 Assignments


Mathematics Placement Test

bull Students who are planning to do Math 1000 for the first time must write theMathematics Placement Test

bull The test will be administered for the last time this semester on Sep 11(Tuesday) at 1900 in HH-3017

bull Failure to write the MPT will result in deregistration from the course

bull The mark required to qualify for Math 1000 is 85 with Academic math 3204 or75 with Advanced math 3205

bull A synopsis of the MPT is available on the shelf (adjacent to HH-3009) acrossthe corridor from the General Office or online athttpwwwmuncamathmptsynopsis


Assignments

bull Homework assignment must be handed in no later than 1400 on the due dateThey may be submitted in class or placed in the marking box Late assignmentwill NOT be marked

bull Assignments should be submitted on loose leaf 85rdquotimes 11rdquo paper stapled inthe upper left corner written on one side only with problems done in orderThey should not be placed in folders or any other container Assignmentsshould include a cover page with your name and student number the name ofthe course and the assignment number

bull If you are unable to submit an assignment due to illness bereavement or otheracceptable reason please provide me with appropriate documentation (egdoctorrsquos note death notice) and I will average together your remainingassignments to create a mark out of 75


WebAssign

bull Website httpwwwmathmuncawebassign

bull The above website contains complete instructions for signing up for WebAssignand you should read it fully

bull In order to fully activate your WebAssign account you will need to purchase aWebAssign access code from the University Bookstore by September 18th

bull You MUST use your MUN email address when signing up for WebAssign

bull Should you have further questions with WebAssign please contact theWebAssign administrator Dr Sullivan (shannonmunca)


Tests and exam

bull No notes textbooks calculators or any electronic communications device arepermitted during any test or final examination

bull There will be no make-up for a missed in-class test Upon presentation ofdocumentation of a valid excuse the corresponding percentage of the finalmark will be added to the final exam With no presentation of suchdocumentation a grade of zero will be entered for the missed test

bull Tentative dates for three in-class tests (will be confirmed in class)

1 Sep 28 Friday2 Oct 19 Friday3 Nov 09 Friday


Supplementary exams

The Department of Mathematics and Statistics offers supplementary examinationsto students satisfying all of the following

bull the student is in clear academic standing

bull the student has a passing term mark and

bull the studentrsquos final grade is 45 to 49


Whenever you need help

bull Math Help Center (HH-3015) stating from Sep 10 (Monday)

1000-1600 Monday Tuesday Wednesday and Thursday0900-1300 Friday

bull Office hour (HH-2016) 1000-1130 Monday Tuesday and Wednesday

bull Email xswang4gmailcom

bull Office phone number 864-4321

bull I am always willing to help you -)


Niagara Falls ON


St John River NB


Difference between the falls and the river

bull The river has no jump It is continuous

bull The falls has a jump It is discontinuous

bull Question how to distinguish between continuity and discontinuity

bull Answer with the aid of limits

bull Remark Limits help us define continuity


Continuity and discontinuity

The slides are continuous The stairs are discontinuous


Discontinuity the unfinished bridge

The function is not defined at the discontinuous point


Discontinuity the broken bridge

The limits from both sides do not match


Numerical investigation of f(x) = sin xx

x f(x) = sin xx x f(x) = sin x

x

1 0841470985 minus1 084147098501 0998334166 minus01 0998334166001 0999983333 minus001 09999833330001 0999999833 minus0001 0999999833

f(x) = sin xx approaches 1 when x is close (but not equal) to 0


Graphical investigation of f(x) = sin xx

x

fHxL=

sinHxLx

1



Definition of limits

bull f(x) is defined for all x near c but not necessarily at c itself

bull |f(x)minus L| becomes arbitrarily small when x is any number sufficiently close(but not equal) to c

bull We say that the limit of f(x) as x approaches c is equal to L

bull We writelimxrarrc

f(x) = L

bull We also say that f(x)rarr L (f(x) approaches or converges to L) as xrarr c (xtends to c)


Limits of two simple functions

For any constant k and c we have

limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c


Numerical investigation of limxrarr1

xminus1radicxminus1

x xminus1radicxminus1 x xminus1radic

xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1

xminus 1radicxminus 1

= 2


Graphical investigation of limxrarr1

xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1

xminus1radicxminus1 = 2



x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x

x exminus1x x exminus1

x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x

1 0841471 minus1 minus084147101 minus05440211 minus01 05440211001 minus05063656 minus001 050636560001 082687954 minus0001 minus082687954

limxrarr0

sin 1x does not exist



sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits

bull Left-hand limit limxrarrcminus

f(x) = L if f(x) converges to L as x approaches c from

left-hand side

bull Right-hand limit limxrarrc+

f(x) = L if f(x) converges to L as x approaches c

from right-hand side

bull Theorem limxrarrc

f(x) = L if and only if both limxrarrcminus

f(x) = L and limxrarrc+

f(x) = L

are satisfied


The absolute value function f(x) = |x|

x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3



x |x| minusx x |x| minusx1 1 minus1 minus1 1 12 2 minus2 minus2 2 23 3 minus3 minus3 3 3

When x gt 0 we have |x| = x When x lt 0 we have |x| = minusx


|x| and piecewise function

bull When x gt 0 we replace |x| by x So

2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1

bull When x lt 0 we replace |x| by minusx So

2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1

bull The function f(x) = 2x+|x|x+2|x| is a piecewise function

2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0


When the limit is infinitybull If f(x) increases without bound as xrarr c then we write lim

xrarrcf(x) =infin

bull If f(x) tends to minusinfin (ie f(x) becomes negative and |f(x)| rarr infin) as xrarr cthen we write lim

xrarrcf(x) = minusinfin

bull If f(x)rarrinfin as x approaches c from the left-hand side (xrarr cminus) then wewrite lim

xrarrcminusf(x) =infin

bull If f(x)rarrinfin as x approaches c from the right-hand side (xrarr c+) then wewrite lim

xrarrc+f(x) =infin

bull If f(x)rarr minusinfin as xrarr cminus then we write limxrarrcminus

f(x) = minusinfin

bull If f(x)rarr minusinfin as xrarr c+ then we write limxrarrc+

f(x) = minusinfin


The function f(x) = 1x

x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist


The function f(x) = 1x2

x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin


The function f(x) = lnx

x

fHxL=lnHxL

limxrarr0minus

lnx does NOT exist limxrarr0+

lnx = minusinfin limxrarr0

lnx does NOT exist


The notation infin

bull The notation infin looks like a sideways 8

limxrarr8+

1

xminus 8=infin

bull Good mathematical skills

limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws

Assume that limxrarrc

f(x) and limxrarrc

g(x) exist Then

(i) Sum Lawlimxrarrc

(f(x) + g(x)) = limxrarrc

f(x) + limxrarrc

g(x)

(ii) Constant Multiple Law For any number k

limxrarrc

kf(x) = k limxrarrc

f(x)

(iii) Product Lawlimxrarrc

(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))

(iv) Quotient Law If limxrarrc

g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws

bull The Sum and Product Laws are valid for any finite number of functions Forexample assume that lim

xrarrcf1(x) lim

xrarrcf2(x) and lim

xrarrcf3(x) exist Then

limxrarrc

(f1(x) + f2(x) + f3(x)) = limxrarrc

f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))

bull The Sum Law and Constant Multiple Law imply Difference Law Assume thatlimxrarrc

f(x) and limxrarrc

g(x) exist Then

limxrarrc

(f(x)minus g(x)) = limxrarrc

f(x) + limxrarrc

(minusg(x)) = limxrarrc

f(x)minus limxrarrc

g(x)

bull To apply the Limit Laws we need to assume limxrarrc

f(x) and limxrarrc

g(x) exist

bull We have to do algebraic transformations before applying the Limit Laws


The squeeze theorem

x

y uHxL

lHxL

fHxL

f(x) is squeezed at x = 0 by l(x) and u(x)


The squeeze theorem

x

y uHxL

lHxL

fHxL

f(x) is NOT squeezed at x = 0 by l(x) and u(x)


The squeeze theorem

If f(x) is squeezed at x = c by l(x) and u(x) namely

1 l(x) le f(x) le u(x) for all x close (but not equal) to c2 lim

xrarrcl(x) = lim

xrarrcu(x) = L

Then the limit limxrarrc

f(x) exists and limxrarrc

f(x) = L

The meatvegetable is squeezed by the two breads


The squeeze theorem limxrarr0

x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x

f(x) = x sin 1x is squeezed at x = 0 by l(x) = minus|x| and u(x) = |x|


Three techniques for evaluating limits and two formulas

bull Three techniques

1 Algebraic Transformation2 The Squeeze Theorem3 Change of Variable

bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0


Limits at infinity and asymptotes

bull If |f(x)minus L| becomes arbitrarily small (f(x)rarr L) as x increases withoutbound (xrarrinfin) then we write

limxrarrinfin

f(x) = L

bull If |f(x)minus L| becomes arbitrarily small (f(x)rarr L) as x decreases withoutbound (xrarr minusinfin) then we write

limxrarrminusinfin

f(x) = L

bull The horizontal line y = L is called a horizontal asymptote of f(x) if

limxrarrinfin

f(x) = L or limxrarrminusinfin

f(x) = L

bull The vertical line x = c is called a vertical asymptote of f(x) if the left-hand orright-hand limit of f(x) at x = c is equal to infin or minusinfin


Limits at infinity

bull For any a gt 0 we have

limxrarrinfin

xminusa = 0 and limxrarrinfin

xa =infin

bull Three properties

1 If limxrarrinfin

f(x) =infin and limxrarrinfin

g(x) = L then limxrarrinfin

(f(x)plusmn g(x)) =infin

2 If limxrarrinfin


g(x) =infin then limxrarrinfin

(f(x) + g(x)) =infin

3 If limxrarrinfin


g(x) = L gt 0 then limxrarrinfin

(f(x)g(x)) =infin

If limxrarrinfin


g(x) = L lt 0 then limxrarrinfin

(f(x)g(x)) = minusinfin


x2 runs faster than x as xrarrinfin

x2 is the hare (rabbit) x is the tortoise (turtle) x2 runs faster than x as xrarrinfin

limxrarrinfin

(x2 minus x) =infin


Limits of polynomials at infinity

bull Transform the differencesum into a product

limxrarrinfin

(x2 minus x) = limxrarrinfin

x2(1minus xminus1) =infin

bull The leading term of a polynomial dominates when xrarrinfin namely if n ge 1and an 6= 0 then

limxrarrinfin

(anxn + anminus1x

nminus1 + middot middot middot+ a0) = limxrarrinfin

xn(an + anminus1xminus1 + middot middot middot+ a0x

minusn)

= limxrarrinfin

anxn

=

infin for an gt 0

minusinfin for an lt 0


Limits of rational functions and limits at minusinfin

bull Let an 6= 0 and bm 6= 0 then

limxrarrinfin

anxn + anminus1x

nminus1 + middot middot middot+ a0bmxm + bmminus1xmminus1 + middot middot middot b0

= limxrarrinfin


minusn)

xm(bm + bmminus1xminus1 + middot middot middot b0xminusm)

= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm

bull If xrarr minusinfin then we make a change of variable y = minusx (replace xrarr minusinfin byy rarrinfin and x by minusy)

limxrarrminusinfin

f(x) = limyrarrinfin

f(minusy)


Horizontal and vertical asymptotes

bull The horizontal asymptotes come from the limits at infinity (if exist)

bull The vertical asymptotes of a rational function come from the point when thedenominator of this rational function becomes zero

bull Steps in finding asymptotes of a rational function

1 Evaluate the limits at infin and minusinfin to obtain horizontal asymptotes2 Find the points when the denominator of this rational function becomes zero3 For each point found in the previous step evaluate the one-sided limits at

this point If one of the limits is infin or minusinfin then we obtain a verticalasymptote otherwise both one-sided limits are finite and this point doesNOT give a vertical asymptote


Continuity

Let f(x) be defined for x near c AND x = c We say f is continuous at x = c if

limxrarrc

f(x) = f(c)

Otherwise we say f is discontinuous at x = c

1 We say f has a removable discontinuity at x = c if

limxrarrc

f(x) 6= f(c)

2 We say f has a jump discontinuity at x = c if

limxrarrcminus

f(x) 6= limxrarrc+

f(x)

3 We say f has an infinite discontinuity at x = c if either or both theon-sided limit is infin or minusinfin


One-sided continuity and laws of continuity

bull f(x) is called left-continuous if

limxrarrcminus

f(x) = f(c)

f(x) is called right-continuous if

limxrarrc+

f(x) = f(c)

bull (Laws of Continuity) If f(x) and g(x) are continuous at x = c Thenf(x)plusmn g(x) kf(x) f(x)g(x) are continuous at x = c If further g(c) 6= 0then f(x)g(x) is continuous at x = c


An example

x

y

-1 1 2

-1

1

f(x) is continuous at x = 1 and discontinuous at x = minus1 0 2f(x) has an infinite discontinuity at x = minus1

a jump discontinuity at x = 0and a removable discontinuity at x = 2

f(x) is left-continuous (but NOT right-continuous) at x = 0


Basic functions inverse function and composite function

bull (Polynomial and rational functions) Let P (x) and Q(x) be polynomials Then

1 P (x) is continuous on the real line2 P (x)Q(x) is continuous at x = c if Q(c) 6= 0

bull (Basic functions)

1 f(x) = sinx and f(x) = cosx are continuous on the real line2 For b gt 0 f(x) = bx is continuous on the real line3 For b gt 0 and b 6= 1 f(x) = logb x is continuous for x gt 04 If n is a rational number then f(x) = x1n is continuous on its domain

bull (Inverse function) If f(x) is continuous on an interval I with range R and ifthe inverse fminus1(x) exists then fminus1(x) is continuous on R

bull (Composite function) If g(x) is continuous at x = c and if f(y) is continuousat y = g(c) then f(g(x)) is continuous at x = c


Rates of change

bull The average rate of change of a function y = f(x) over an interval [x0 x1] isdefined as

∆y

∆x=f(x1)minus f(x0)

x1 minus x0

bull The instantaneous rate of change of a function y = f(x) at x = x0 is definedas

limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0

f(x0 + h)minus f(x0)

h

bull The average rate of change is equal to the slope of the secant line through(x0 f(x0)) and (x1 f(x1))

bull The instantaneous rate of change is equal to the slope of the tangent line atx = x0


Rates of change

(Loading Video)

The secant line (red) approaches the tangent line (green)The average rate of change approaches the instantaneous rate of change


tangentavi
Media File (videoavi)

Rates of change bubble gum



bull The volume of the bubble is

V (r) =4

3πr3

bull The average ROC of bubble volume when the radius increases from r0 to r1 is

V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)

bull The instantaneous ROC of bubble volume at r = r0 is

limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20


Rates of change throwing a ball up in the air



bull The height of the ball at time t is given by the displacement function

s(t) = 16tminus 16t2

bull The average ROC of height (average velocity) over [t0 t1] is

s(t1)minus s(t0)t1 minus t0

= 16minus 16(t1 + t0)

bull The instantaneous ROC of height (instantaneous velocity) at t = t0 is

limt1rarrt0


= 16minus 32t0


Derivative and tangent line

bull The derivative of a function f(x) at x = a is the limit of the differencequotient (if exists)

f prime(a) = limxrarra

f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h

If f prime(a) exists then we say that f is differentiable at x = a

bull The tangent line to y = f(x) at x = a has the slope f prime(a) and the equation inpoint-slope form

y minus f(a) = f prime(a)(xminus a)

bull We say y = f(x) is differentiable in (a b) if f prime(x) exists for all x in (a b) Inthis case we view yprime = f prime(x) as a function defined on (a b)


Notations

bull Let y = f(x) then its derivative function is denoted by

yprime = f prime(x) =dy

dx=df

dx=

d

dx(f(x))

For example if y = f(x) = xminus1 then


dx=df

dx=

d

dx(xminus1) = minusxminus2

bull If y = f(x) is differentiable at x = a then we write

yprime(a) = f prime(a) =dy

dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a

For example if y = f(x) = xminus1 and a = 1 then

yprime(1) = f prime(1) =dy

dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems

Let f(x) and g(x) be differentiable functions

bull The Power Rule(xa)prime = axaminus1

bull The Sum Rule(f plusmn g)prime = f prime plusmn gprime

bull The Constant Multiple Rule(cf)prime = cf prime

bull If f is differentiable at x = c then f is continuous at x = c

bull Remark If f is continuous at x = c It is NOT necessarily that f isdifferentiable at x = c (Counter example f(x) = |x| and c = 0)


The product and quotient rules higher derivatives

bull The Product Rule(fg)prime = f primeg + fgprime

bull The Quotient Rule (f

g

)prime=f primeg minus fgprime

g2

bull Higher derivatives let y = f(x) then


dx=df

dx=

d

dx(f(x))

yprimeprime = f primeprime(x) =d

dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))

)yprimeprimeprime = f primeprimeprime(x) = y(3) = f (3)(x) =

d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))

))Lecture Notes for Math 1000 First Previous Next Last 69

Derivatives of trigonometric and exponential functions

bull Three basic formulas

d

dx(sinx) = cosx

d

dx(cosx) = minus sinx and

d

dx(ex) = ex

bull Express everything in terms of sinx and cosx and then apply derivative rules

d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1

sin2 x= minus csc2 x

d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx

sin2 x= minus cscx cotx


The chain rule

bull Let u = g(x) be differentiable at x = x0 and y = f(u) be differentiable atu = u0 = g(x0) then y = f(g(x)) is differentiable at x = x0 and

dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0

bull We can also write

dy

dx=dy

du

du

dxor

d

dxf(g(x)) = f prime(g(x))gprime(x) or

d

dxf(u) = f prime(u)

du

dx


The chain rule (special applications)

bull If g(x) is differentiable then

d

dx(g(x))a = a(g(x))aminus1gprime(x) and

d

dxeg(x) = gprime(x)eg(x)

bull If f(u) is differentiable then

d

dxf(kx+ b) = kf prime(kx+ b)

Especially if f(u) = eu then

(ekx+b)prime = kekx+b

bull If b gt 0 then(bx)prime = (ln b)bx


Implicit differentiation

bull Given an implicit function

y4 + xy = x3 minus x+ 2

bull Take derivative with respect to x on both side of the equation

d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)

bull Apply the chain rule and other derivative rules to obtain

4y3dy

dx+ y + x

dy

dx= 3x2 minus 1

bull Solvedydx from the above equation

dy

dx=

3x2 minus 1minus y4y3 + x


Derivative of inverse function

bull If y = f(x) is differentiable at x = x0 and f prime(x0) 6= 0 Then then inversefunction x = g(y) = fminus1(y) is differentiable at y = y0 = f(x0) and

gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))

bull Inverse trigonometric functions

(sinminus1 y)prime =1

(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2

(cosminus1 y)prime =1

(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2


y = sinx and y = sinminus1 x

x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2

]sinminusminusminusminusminusminussinminus1

[minus1 1]


y = cosx and y = cosminus1 x

x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2

[0 π]cosminusminusminusminusminusminuscosminus1

[minus1 1]


y = tanx and y = tanminus1 x

x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2

)tanminusminusminusminusminusminustanminus1

(minusinfininfin)


y = cotx and y = cotminus1 x

x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2

(0 π)cotminusminusminusminusminusminuscotminus1

(minusinfininfin)


y = cscx and y = cscminus1 x

x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2

]cscminusminusminusminusminusminuscscminus1

(minusinfinminus1] cup [1infin)


y = secx and y = secminus1 x

x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]

secminusminusminusminusminusminussecminus1



Derivatives of inverse trigonometric functions


(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2

(tanminus1 y)prime =1

(tanx)prime= cos2 x =

cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2

(cotminus1 y)prime =1

(cotx)prime= minus sin2 x = minus sin2 x

sin2 x+ cos2 x= minus 1

1 + cot2 x= minus 1

1 + y2

(secminus1 y)prime =1

(secx)prime=

cos2 x

sinx=

cos2 xradic1minus cos2 x

=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1

(cscminus1 y)prime =1

(cscx)prime=minus sin2 x

cosx=

minus sin2 xradic1minus sin2 x

=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1


Derivatives of logarithmic functions and hyperbolic functions

bull Logarithmic functions

(log x)prime =1

xand (log g(x))prime =

gprime(x)

g(x)

bull Hyperbolic sine and cosine functions

sinhx =ex minus eminusx

2and coshx =

ex + eminusx

2

bull Identitycosh2xminus sinh2x = 1

bull Derivatives(sinhx)prime = coshx and (coshx)prime = sinhx


Derivatives of hyperbolic functions

bull Hyperbolic tangent and cotangent functions

tanhx =sinhx

coshx=ex minus eminusx

ex + eminusxand cothx =

coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=

(sinhx)prime(coshx)minus (sinhx)(coshx)prime

cosh2x

=(coshx)(coshx)minus (sinhx)(sinhx)

cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=

(coshx)prime(sinhx)minus (coshx)(sinhx)prime

sinh2x

=(sinhx)(sinhx)minus (coshx)(coshx)

sinh2x=minus1

sinh2x= minuscsch2x



bull Hyperbolic secant and cosecant functions

sechx =1

coshx=

2

ex + eminusxand cschx =

1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=

(1)prime(coshx)minus (1)(coshx)prime

cosh2x=minussinhx

cosh2x= minustanhxsechx

(cschx)prime =

(1

sinhx

)prime=

(1)prime(sinhx)minus (1)(sinhx)prime

sinh2x=minuscoshx

sinh2x= minus cothxcschx

bull Express every hyperbolic function in terms of sinh and cosh


Absolute extrema

bull Let f(x) be defined on an interval I and let c isin I We say

1 f(c) is an absolute maximum if f(x) le f(c) for all x isin I2 f(c) is an absolute minimum if f(x) ge f(c) for all x isin I

bull If f(x) is continuous on the closed and bounded interval I = [a b] then f(x)has absolute extrema (maximum and minimum) on I

bull If f prime(c) is not defined or f prime(c) = 0 then c is called a critical point of f(x)

bull Let f(x) be continuous on the closed and bounded interval I = [a b] If f(c) isan absolute extremum (maximum or minimum) of f(x) on I then c is either acritical point or an end point


Relative extrema

bull Let f(x) be defined near x = c and x = c is not an end point

1 We say f(c) is a relative maximum if f(x) le f(c) for all x close to c2 We say f(c) is a relative minimum if f(x) ge f(c) for all x close to c

bull Let f(x) be defined on an interval I

1 We say f(x) is a increasing on I if f(x1) lt f(x2) for all x1 lt x22 We say f(x) is a decreasing on I if f(x1) gt f(x2) for all x1 lt x2

bull Assume f prime(x) exists for all x isin I

1 If f prime(x) gt 0 for all x isin I then f(x) is a increasing on I2 If f prime(x) lt 0 for all x isin I then f(x) is a decreasing on I

bull (First Derivative Test for Critical Points) Let f(x) be defined near x = cAssume f prime(x) exists for all x near c but not necessarily at c itself

1 If f prime(x) changes from + to minus at x = c then f(c) is a relative maximum2 If f prime(x) changes from minus to + at x = c then f(c) is a relative minimum


Relative extrema points of inflection and derivative tests

bull Assume f primeprime(x) exists for all x isin I

1 We say f(x) is a concave up if f primeprime(x) gt 0 for all x isin I2 We say f(x) is a concave down if f primeprime(x) lt 0 for all x isin I3 If f primeprime(x) changes sign at x = c we say x = c is a point of inflection



bull (Second Derivative Test for Critical Points) Assume f prime(c) = 0

1 If f primeprime(c) lt 0 then f(c) is a relative maximum2 If f primeprime(c) gt 0 then f(c) is a relative minimum

bull If f(c) is a relative extremum (maximum or minimum) then x = c is a criticalpoint namely either f prime(c) does not exist or f prime(c) = 0


Curve sketching

Step 1 Find the xminus and yminus intercepts

Step 2 Find the horizontal and vertical asymptotes

Step 3 Determine the intervals on which the function is increasing or decreasingand classify the relative extrema

Step 4 Determine the intervals on which the function is concave up or concavedown and identify the points of inflection

Step 5 Sketch the curve


LrsquoHopitalrsquos rule

bull Assume f prime(x) and gprime(x) exist for all x near a and gprime(x) 6= 0 for x near but notequal to a If f(a) = g(a) = 0 then

limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)

provided that the limit on the right exists

bull If limxrarra

f(x) = plusmninfin and limxrarra

g(x) = plusmninfin then LrsquoHopitalrsquos rule also applies

Furthermore the limits may be taken as one-sided limits

bull Assume f prime(x) and gprime(x) exist for large x and gprime(x) 6= 0 for all large x If bothlimxrarrinfin

f(x) and limxrarrinfin

g(x) are zero (or infinity) then

limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)

provided that the limit on the right exists Similar results hold when xrarr minusinfin


Self-introduction

bull Name Xiang-Sheng Wang

bull Position Teaching postdoctoral fellow

bull PhD received 2009

bull Previous affiliation York University

bull Teaching experience

1 Math 1013 (Applied Calculus I)2 Math 1190 (Introduction to Sets and Logic)3 Math 1540 (Introductory Mathematics for Economists II)4 Math 6378 (Applied Delay Differential Equations graduate course)
























Outline















Assignments





WebAssign







Tests and exam






Supplementary exams














Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)

























Outline















Assignments





WebAssign







Tests and exam






Supplementary exams














Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




















Outline















Assignments





WebAssign







Tests and exam






Supplementary exams














Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)















Outline















Assignments





WebAssign







Tests and exam






Supplementary exams














Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)









Outline















Assignments





WebAssign







Tests and exam






Supplementary exams














Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Outline















Assignments





WebAssign







Tests and exam






Supplementary exams














Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)










Assignments





WebAssign







Tests and exam






Supplementary exams














Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Assignments





WebAssign







Tests and exam






Supplementary exams














Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



WebAssign







Tests and exam






Supplementary exams














Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Tests and exam






Supplementary exams














Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Supplementary exams














Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)











Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Niagara Falls ON


St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



St John River NB




















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





















x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)














x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)











x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)








x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





x





x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x

fHxL=

sinHxLx

1








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)








f(x) = L





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





limxrarrc

k = k and limxrarrc

x = c

x

f HxL=k

k

c

x

f HxL=x

c

c



xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




xminus1radicxminus1


xminus1

0 1 2 241409 19487 11 20488099 1994987 101 20049880999 199949987 1001 200049988

limxrarr1


= 2



xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




xminus1radicxminus1

x

fHxL=x - 1

x - 1

2

1

limxrarr1




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x2

x x2 x x2

0 0 2 409 081 11 121099 098 101 102010999 0998 1001 1002001

limxrarr1

x2 = 1



x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x2

x

fHxL=x2

1

1

limxrarr1

x2 = 1



exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




exminus1x


x


limxrarr0

ex minus 1

x= 1



exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




exminus1x

x

fHxL=atildex - 1

x

1

limxrarr0

exminus1x = 1



cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




cosx

x cosx x cosx


limxrarr0

cosx = 1



cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




cosx

x

fHxL=cosHxL

1

limxrarr0

cosx = 1



sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




sin 1x

x sin 1x x sin 1

x


limxrarr0




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




sin 1x

x

fHxL=sin1

x

limxrarr0



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



One-sided limits



left-hand side







f(x) = L

are satisfied



x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x

fHxL= xcurren

-1-2-3 1 2 3

1

2

3








2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)









2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





2x+ |x|x+ 2|x|

=2x+x

x+2x=

3x

3x= 1


2x+ |x|x+ 2|x|

=2xminusxxminus2x

=x

minusx= minus1


2x+ |x|x+ 2|x|

=

1 for x gt 0

minus1 for x lt 0



xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




xrarrcf(x) =infin






xrarrc+f(x) =infin


f(x) = minusinfin


f(x) = minusinfin



x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x

fHxL=

1

x

limxrarr0minus

1x = minusinfin lim

xrarr0+

1x =infin lim

xrarr0

1x does NOT exist



x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x

fHxL=

1

x2

limxrarr0minus

1x2 =infin lim

xrarr0+

1x2 =infin lim

xrarr0

1x2 =infin



x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x

fHxL=lnHxL

limxrarr0minus



lnx does NOT exist


The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



The notation infin


limxrarr8+

1

xminus 8=infin


limxrarr2+

1

xminus 2=

2 times

limxrarr2+

1

xminus 2= 2 times

limxrarr2+

1

xminus 2=infin X


Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Basic limit laws


f(x) and limxrarrc

g(x) exist Then



f(x) + limxrarrc

g(x)


limxrarrc

kf(x) = k limxrarrc

f(x)


(f(x)g(x)) =(

limxrarrc

f(x))(

limxrarrc

g(x))


g(x) 6= 0 then

limxrarrc

f(x)

g(x)=

limxrarrc

f(x)

limxrarrc

g(x)


Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Basic limit laws


xrarrcf1(x) lim

xrarrcf2(x) and lim


limxrarrc


f1(x) + limxrarrc

f2(x) + limxrarrc

f3(x)

limxrarrc

(f1(x)f2(x)f3(x)) =(

limxrarrc

f1(x))(

limxrarrc

f2(x))(

limxrarrc

f3(x))


f(x) and limxrarrc

g(x) exist Then

limxrarrc


f(x) + limxrarrc


f(x)minus limxrarrc

g(x)


f(x) and limxrarrc

g(x) exist



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



The squeeze theorem

x

y uHxL

lHxL

fHxL



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



The squeeze theorem



xrarrcl(x) = lim

xrarrcu(x) = L



f(x) = L




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x sin 1x = 0

x

y

uHxL=EgravexEgrave

lHxL=-EgravexEgrave

fHxL=x sin1

x






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)






bull Two formulas

limxrarr0

sinx

x= 1 and lim

xrarr0

1minus cosx

x= 0




limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





limxrarrinfin

f(x) = L


limxrarrminusinfin

f(x) = L


limxrarrinfin


f(x) = L



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Limits at infinity


limxrarrinfin


xa =infin


1 If limxrarrinfin




2 If limxrarrinfin




3 If limxrarrinfin



(f(x)g(x)) =infin

If limxrarrinfin







limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





limxrarrinfin

(x2 minus x) =infin




limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





limxrarrinfin




limxrarrinfin

(anxn + anminus1x



minusn)

= limxrarrinfin

anxn

=

infin for an gt 0





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





limxrarrinfin

anxn + anminus1x


= limxrarrinfin


minusn)


= limxrarrinfin

anxn

bmxm

=anbm

limxrarrinfin

xnminusm


limxrarrminusinfin


f(minusy)









Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)










Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Continuity


limxrarrc

f(x) = f(c)



limxrarrc

f(x) 6= f(c)


limxrarrcminus

f(x) 6= limxrarrc+

f(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





limxrarrcminus

f(x) = f(c)


limxrarrc+

f(x) = f(c)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



An example

x

y

-1 1 2

-1

1













Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)











Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Rates of change


∆y


x1 minus x0


limx1rarrx0

f(x1)minus f(x0)

x1 minus x0= lim

hrarr0


h




Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Rates of change

(Loading Video)



tangentavi





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)







V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





V (r) =4

3πr3


V (r1)minus V (r0)

r1 minus r0=

4

3π(r21 + r1r0 + r20)


limr1rarrr0

V (r1)minus V (r0)

r1 minus r0= 4πr20









= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)










= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)








= 16minus 16(t1 + t0)


limt1rarrt0


= 16minus 32t0





f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)






f(x)minus f(a)

xminus a= lim

hrarr0

f(a+ h)minus f(a)

h






Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Notations



dx=df

dx=

d

dx(f(x))



dx=df

dx=

d




dx

∣∣∣∣x=a

=df

dx

∣∣∣∣x=a



dx

∣∣∣∣x=1

=df

dx

∣∣∣∣x=1

=(minusxminus2

) ∣∣∣∣x=1

= minus1


Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Theorems











g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)






g


g2



dx=df

dx=

d

dx(f(x))


dx

(dy

dx

)=

d

dx

(dy

dx

)=d2y

dx2=d2f

dx2=

d

dx

(d

dx(f(x))


d3y

dx3=d3f

dx3=

d

dx

(d

dx

(d

dx(f(x))




d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





d

dx(sinx) = cosx

d


d

dx(ex) = ex


d

dx(tanx) =

d

dx

(sinx

cosx

)=

1

cos2 x= sec2 x

d

dx(cotx) =

d

dx

(cosx

sinx

)= minus 1


d

dx(secx) =

d

dx

(1

cosx

)=

sinx

cos2 x= secx tanx

d

dx(cscx) =

d

dx

(1

sinx

)= minus cosx



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



The chain rule


dy

dx

∣∣∣∣x=x0

=dy

du

∣∣∣∣u=u0=g(x0)

times dudx

∣∣∣∣x=x0


dy

dx=dy

du

du

dxor

d


d

dxf(u) = f prime(u)

du

dx




d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





d


d



d










d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)







d

dx(y4 + xy) =

d

dx(x3 minus x+ 2)


4y3dy

dx+ y + x

dy

dx= 3x2 minus 1


dy

dx=





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





gprime(y0) =1

f prime(x0)=

1

f prime(g(y0))



(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x

y=sinHxL

-Π 2

-1

1

Π 2x

y=sin-1HxL

-Π 2

-1

1

Π 2

[minusπ

2π

2


[minus1 1]



x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x

y=cosHxL

Π

-1

1

Π 2

x

y=cos-1HxL

Π

-1 1

Π 2


[minus1 1]



x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x

y=tanHxL

-Π 2 Π 2 x

y=tan-1HxL

-Π 2

Π 2

(minusπ

2π

2


(minusinfininfin)



x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x

y=cotHxL

ΠΠ 2

x

y=cot-1HxLΠ

Π 2


(minusinfininfin)



x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x

y=cscHxL

-Π 2

-1

1

Π 2 x

y=csc-1HxL

-Π 2

-11

Π 2

[minusπ

2 0)cup(

0π

2





x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)




x

y=secHxL

Π

-1

1

Π 2

x

y=sec-1HxLΠ

-1 1

Π 2

[0π

2

)cup(π

2 π]






(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





(sinx)prime=

1

cosx=

1radic1minus sin2 x

=1radic

1minus y2


(cosx)prime=

1

minus sinx=

1

minusradic

1minus cos2 x=

1

minusradic

1minus y2



cos2 x

cos2 x+ sin2 x=

1

1 + tan2 x=

1

1 + y2




1 + cot2 x= minus 1

1 + y2


(secx)prime=

cos2 x

sinx=


=1

y2radic

1minus (1y)2=

1

|y|radicy2 minus 1



cosx=


=minus1

y2radic

1minus (1y)2=

minus1

|y|radicy2 minus 1




(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





(log x)prime =1


gprime(x)

g(x)



2and coshx =

ex + eminusx

2






tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





tanhx =sinhx



coshx

sinhx=ex + eminusx

ex minus eminusx

bull Derivatives

(tanhx)prime =

(sinhx

coshx

)prime=


cosh2x


cosh2x=

1

cosh2x= sech2x

(cothx)prime =

(coshx

sinhx

)prime=


sinh2x


sinh2x=minus1

sinh2x= minuscsch2x




sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





sechx =1

coshx=

2


1

sinhx=

2

ex minus eminusx

bull Derivatives

(sechx)prime =

(1

coshx

)prime=


cosh2x=minussinhx


(cschx)prime =

(1

sinhx

)prime=


sinh2x=minuscoshx




Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Absolute extrema







Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Relative extrema



















Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)












Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Curve sketching









limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)





limxrarra

f(x)

g(x)= lim

xrarra

f prime(x)

gprime(x)


bull If limxrarra







limxrarrinfin

f(x)

g(x)= lim

xrarrinfin

f prime(x)

gprime(x)



Lecture Notes for Math 1000 - Memorial University of ...xiangshengw/math1000/math1000.pdf · Explained the material clearly. ... Xiangsheng Wang methods are much better from going

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