CHAPTER 2 – CHAPTER 2 – PART I PART I The Derivative The Derivative Faculty of Science & Technology Faculty of Science & Technology pages : pages :
Feb 20, 2016
CHAPTER 2 – CHAPTER 2 – PART IPART I
The DerivativeThe DerivativeFaculty of Science & TechnologyFaculty of Science & Technology
pages : pages :
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 22
OUTLINEOUTLINEThe Tangent LineThe Tangent LineDefinition of The DerivativeDefinition of The DerivativeRules for Finding DerivativesRules for Finding DerivativesDerivatives of Trigonometric FunctionsDerivatives of Trigonometric FunctionsThe Chain RuleThe Chain RuleHigher-Order DerivativesHigher-Order DerivativesImplicit DifferentiationImplicit DifferentiationApplications of The DerivativeApplications of The Derivative
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 33
Tangent DefinitionTangent DefinitionFrom geometryFrom geometry– a line in the plane of a circlea line in the plane of a circle– intersects in exactly intersects in exactly oneone point point
We wish to enlarge on the idea to include We wish to enlarge on the idea to include tangency to any function, f(x)tangency to any function, f(x)
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 44
Slope of Line Tangent to a Slope of Line Tangent to a CurveCurve
Approximated by secantsApproximated by secants– twotwo points of intersection points of intersection
Let second point get closer and closer to Let second point get closer and closer to desired point of tangencydesired point of tangency
•• •
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 55
Animated Secant LineAnimated Secant Line
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 66
Animated TangentAnimated Tangent
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 77
Slope of Line Tangent to a Slope of Line Tangent to a CurveCurve
Recall the concept of a limit fromRecall the concept of a limit from previous chapter previous chapterUse the limit in this contextUse the limit in this context ••
0 0
0
( ) ( )limx
f x x f xmx
x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 88
The Slope Is a LimitThe Slope Is a LimitConsider f(x) = xConsider f(x) = x33.. Find the tangent at Find the tangent at xx00= 2= 2
Now finish …Now finish …
0
3 3
0
2 3
0
(2 ) (2)lim
(2 ) 2lim
8 12 6( ) ( ) 8lim
x
x
x
f x fmx
xmxx x xm
x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 99
Definition of DerivativeDefinition of DerivativeThe derivative is the The derivative is the formulaformula which which gives the slope of the tangent line at gives the slope of the tangent line at any point any point xx for for ff((xx))
Note: the limit Note: the limit must existmust exist no holeno hole no jumpno jump no poleno pole no sharp cornerno sharp corner
0 0
0
( ) ( )'( ) limh
f x h f xf x
h
A derivative is a limit !
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 1010
Derivative NotationDerivative NotationFor the function For the function yy = = ff((xx))Derivative may be expressed as …Derivative may be expressed as …
'( ) " prime of "
"the derivative of with respect to "
f x f xdy y xdx
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 1111
FindFind f f ’’((xx) ) using definition.using definition.
0
0
0
0 0
[sin( ) 1] [sin 1]'( ) lim
sin( )cos( ) cos( )sin( ) 1 sin 1lim
sin( )(cos( ) 1) cos( )sin( )lim
sin( ) cos( )
cos(
(cos( ) 1) sin( )lim
( ) s
i
)
l
1
m
in
h
h h
h
h
x h xf xh
x h x h xh
x h x h
f x x
h hh
h
x x
xh
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 1212
Differentiability implies Differentiability implies continuity.continuity.
If the graph of a function has a tangent at If the graph of a function has a tangent at point c, then there is no “jump” on the point c, then there is no “jump” on the graph at that point, thus is continuous graph at that point, thus is continuous there.there.
If ' , then is continuous at f c f c
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 1313
A Derivative is a LimitA Derivative is a LimitTherefore, the rules for limits, Therefore, the rules for limits, essentially become the rules for essentially become the rules for derivatives.derivatives.Derivative of a sum/difference is the Derivative of a sum/difference is the sum/difference of the derivatives.sum/difference of the derivatives.Derivative of a product/quotient is the Derivative of a product/quotient is the product/quotient of the derivatives.product/quotient of the derivatives.
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 1414
Basic DerivativesBasic DerivativesConstant FunctionConstant Function– Given f(x) = kGiven f(x) = k– Then f’(x) = 0Then f’(x) = 0
Power FunctionPower Function– Given f(x) = x Given f(x) = x nn
– Then Then 1'( ) nf x n x
( ) 0d kdx
1n nd x n xdx
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 1515
Basic DerivativesBasic Derivatives
Identity FunctionIdentity Function– Given f(x) = xGiven f(x) = x– Then f’(x) = 1Then f’(x) = 1
( ) 1d xdx
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 1616
Basic RulesBasic RulesConstant multipleConstant multipleRuleRule
Sum RuleSum Rule
DifferenceDifferenceRuleRule
( ) ( )d dc f x c f xdx dx
( ) ( ) ( ) ( )d d df x g x f x g xdx dx dx
( ) ( ) ( ) ( )d d df x g x f x g xdx dx dx
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 1717
Try It OutTry It OutDetermine the followingDetermine the following
2 2 3' 2 2
y t ty t
( ) 3 5
'( ) 3
f x x
f x
3
4
( )
'( ) 3
p x x
p x x
3
4
143
( )
11 '( )3
xh xx
h x x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 1818
Product RuleProduct RuleConsider the product of two functionsConsider the product of two functions
It can be shown thatIt can be shown that
In words:In words:– The first function times the derivative of the second The first function times the derivative of the second
plus the second function times the derivative of the plus the second function times the derivative of the firstfirst
( ) ( ) ( )f x h x k x
'( ) ( ) '( ) ( ) '( )f x h x k x k x h x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 1919
Quotient RuleQuotient RuleWhen our function is the quotient of two other When our function is the quotient of two other functions …functions …
The quotient rule specifies the derivativeThe quotient rule specifies the derivative
In words:In words:– The denominator times the derivative of the numerator The denominator times the derivative of the numerator
minus the numerator times the derivative of the minus the numerator times the derivative of the denominator, all divided by the square of the denominatordenominator, all divided by the square of the denominator
( )( )( )
p xf xq x
2
( ) '( ) ( ) '( )'( )( )
q x p x p x q xf xq x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 2020
Just Checking . . .Just Checking . . .Find the derivatives of the given functionsFind the derivatives of the given functions
2
3 2
3
2
2 3 6 3
' 2 3 6 6 3 3p
p x x x
x x x x x
2
2
22
5 7 7 4 2 ( )
5
7 4( )5
xq x
xq x
x
xx x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 2121
Derivatives of Trigonometric Derivatives of Trigonometric FunctionsFunctions
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 2222
Find the derivative of the Find the derivative of the followingfollowing
2
2
2
2
(sec tan csc ) (sec cot )(1)'
sec tan csc sec cot'
sec cot
x x x x x xyx
x x x x x x xyx
x xyx
The Chain RuleThe Chain Rule
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 2424
Solution: The Chain RuleSolution: The Chain RuleGiven y = f (u) and u = g (x)Given y = f (u) and u = g (x)– That is y = f(u) = f ( g(x) )That is y = f(u) = f ( g(x) )
ThenThen
In words:In words:– The derivative of The derivative of y with respect to xy with respect to x is is
the derivative of the derivative of y with respect to uy with respect to u timestimesthe derivative of the derivative of u with respect to xu with respect to x
dy dy dudx du dx
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 2525
Chain RuleChain RuleExampleExample – given – given– Then andThen and
34 28 3y x x 3( )y f u u 4 28 3u x x
dy dy dudx du dx
34 2 3'( ) 3 8 3 32 6f x x x x x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 2626
Other ExampleOther ExampleGiven:Given:y = (6xy = (6x33 – 4x + 7) – 4x + 7)33
Then u(x) = 6xThen u(x) = 6x33 – 4x + 7 – 4x + 7and f(u) = uand f(u) = u33
ThusThusf’(x) = 3(6xf’(x) = 3(6x33 – 4x + 7) – 4x + 7)22(18x(18x22 – 4) – 4)
dydu
dudx
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 2727
Find the derivativeFind the derivativeNote this is the composition of 3 functions, Note this is the composition of 3 functions, therefore there will be 3 “pieces” to the chain.therefore there will be 3 “pieces” to the chain.
3
3 5
3 4
2
5
' cos[( csc ) ]
[5( csc ) ]
(3 csc c
sin[( csc
)
) ]
ot
y x x
x x
x x
y x
x
x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 2828
Derivatives of Exponential Derivatives of Exponential FunctionsFunctions
ConclusionConclusion
When y = aWhen y = ag(x)g(x)
– Use chain ruleUse chain rule
Similarly for y = eSimilarly for y = eg(x)g(x)
ln( )x xxD a a a
( )ln '( )g x
dy dy dudx du dx
a a g x
( ) ( ) '( )g x g xxD e e g x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 2929
Derivative of the Log Derivative of the Log FunctionFunction
For the natural logarithm For the natural logarithm ln(x)ln(x)
–
For the log of a different base logFor the log of a different base logaa(x)(x)
–
1lnxD xx
1log
lnx aD xa x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 3030
What About ln(-x) ?What About ln(-x) ?Consider it a compound functionConsider it a compound function
Apply the chain ruleApply the chain rule
Thus we see Thus we see
( ) ln( ) ( )( ( ))
f x x g x xy f g x
1 ( ) 1 1ln( ) 1xd xD x
x dx x x
ln( ) ln( )x xD x D x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 3131
Differentiate each of the Differentiate each of the following functions.following functions.
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 3232
Higher-Order DerivativesHigher-Order Derivativesf’’ = 2f’’ = 2ndnd derivative derivativef’’’ = 3f’’’ = 3rdrd derivative derivativef’’’’ = 4f’’’’ = 4thth derivative, etc… derivative, etc…
The 2The 2ndnd derivative is the derivative of derivative is the derivative of the 1the 1stst derivative. derivative.
The 3The 3rdrd derivative is the derivative of derivative is the derivative of the 2the 2ndnd derivative, etc… derivative, etc…
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 3333
Alternate NotationAlternate NotationThere is some alternate notation for higher There is some alternate notation for higher order derivatives as well. order derivatives as well. Recall that there was a fractional notation Recall that there was a fractional notation for the first derivative.for the first derivative.
We can extend this to higher order We can extend this to higher order derivatives :derivatives :
… etc.
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 3434
Example :Example :Find the first four derivatives for each of Find the first four derivatives for each of the following.the following.
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 3535
Solutions :Solutions :(a)(a) (b) (b)
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 3636
Implicit DifferentiationImplicit DifferentiationConsider an equation involving Consider an equation involving bothboth x and y: x and y:
This equation This equation implicitlyimplicitly defines a function in x defines a function in xIt could be defined It could be defined exexplicitlyplicitly
2 2 49x y
2 49 ( 7)y x where x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 3737
DifferentiateDifferentiateDifferentiate both sides of the equationDifferentiate both sides of the equation– each termeach term– one at a timeone at a time– use the chain rule for terms containing yuse the chain rule for terms containing y
For we getFor we get
Now solve for dy/dxNow solve for dy/dx
2 2 49x y
2 2 0dyx ydx
dy xdx y
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 3838
Differentiate cont.Differentiate cont.Then gives usThen gives us
We can replace the y in the results with the We can replace the y in the results with the explicit value of y as neededexplicit value of y as neededThis gives usThis gives usthe slope on the the slope on the curve for any curve for any legal value of xlegal value of x
2 2 0dyx ydx
22
dy x xdx y y
2 49
dy xdx x
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 3939
Second DerivativeSecond DerivativeGiven xGiven x2 2 –y–y22 = 49 = 49
y’ =??y’ =??
y’’ =y’’ =
' xyy
2
2 2
'd y y x ydx y
Substitute
KG/ME_SST/SEPT08KG/ME_SST/SEPT08 4040
Find the derivativeFind the derivative
2
2
2
2
cos( ) ( 1) sec ( ) 2
cos( ) cos( ) sec ( ) 2
( cos( ) sec ) 2 cos( )
2 cos( )cos( ) s
sin( ) tan )
ec
( 2dy dyxy x y ydx dx
dy dyxy x y xy ydx dx
dy x xy y y xydxdy y xydx
xy y
xy
x
x y
Applications OfApplications OfThe DerivativeThe Derivative
Refer pages 151-214 : Chapter 3