Chapter 3 The Derivative By: Kristen Whaley. 3.1 Slopes and Rates of Change Average Velocity Instantaneous Velocity Average Rate of Change Instantaneous.

Chapter 3Chapter 3The DerivativeThe Derivative

By: Kristen WhaleyBy: Kristen Whaley

3.13.1Slopes and Rates of ChangeSlopes and Rates of Change

Average VelocityAverage Velocity Instantaneous VelocityInstantaneous VelocityAverage Rate of ChangeAverage Rate of Change Instantaneous Rate of ChangeInstantaneous Rate of Change

Average VelocityAverage VelocityFor an object moving along an s-axis, For an object moving along an s-axis,

with s= f(t), the average velocity of with s= f(t), the average velocity of an object between times tan object between times t00 and t and t1 1 is:is:

V average = f(t1) – f(t0)V average = f(t1) – f(t0)

t1 - t0t1 - t0

Secant Line: the line determined Secant Line: the line determined by two points on a curveby two points on a curve

Instantaneous VelocityInstantaneous Velocity

For an object moving along an s-axis, For an object moving along an s-axis, with s= f(t), the instantaneous with s= f(t), the instantaneous velocity of the object at time tvelocity of the object at time t0 0 is:is:

V V instantaneousinstantaneous = lim f(t1) – f(t0) = lim f(t1) – f(t0)

t1 - t0t1 - t0 tt11tt00

http://www.coolschool.ca/lor/CALC12/unit2/U02L01/averagevelocityvsinstantaneous.swf

Average and Instantaneous Average and Instantaneous Rates of ChangeRates of Change

Slope can be viewed as a rate of change, Slope can be viewed as a rate of change, and can be useful beyond simple velocity and can be useful beyond simple velocity examples.examples.

r r averageaverage = f(x = f(x11) – f(x) – f(x00))

xx11 - x - x0 0

If y= f(x), the average If y= f(x), the average rate of change over the rate of change over the interval [xinterval [x00, x, x11] of y with ] of y with respect to x is:respect to x is:

If y= f(x), the If y= f(x), the instantaneous rate of instantaneous rate of change of y with respect change of y with respect to x at xto x at x00 is: is:

r r inst.inst. = lim f(x1) – f(x0) = lim f(x1) – f(x0)

x1 - x0 x1 - x0 xx11xx00

Examples!!Examples!!1: Find the slope of the graph of f(x)= 1: Find the slope of the graph of f(x)=

xx22+1 at the point x+1 at the point x00 = 2 = 2

We’re looking for the instantaneous rate of change (slope) of f(x) at x= 2

r inst = lim f(x1) – f(x0)

x1 – x0

x1 x0

r inst = lim f(x1) – f(2)

x1 – 2x1 2

r inst = lim (x1)2 +1 – (22 + 1)

x1 – 2x1 2

r inst = lim (x1 – 2) (x1 + 2)

x1 – 2x1 2

r inst = lim (x1 + 2)

x1 2

r inst = 4

Examples!!Examples!!

2:2: During the first 40s of a rocket flight, the During the first 40s of a rocket flight, the rocket is propelled straight up so that in rocket is propelled straight up so that in t seconds it reaches a height of s=5tt seconds it reaches a height of s=5t33 ft. ft.

How high does the rocket travel? How high does the rocket travel? What is the average velocity of the What is the average velocity of the

rocket during the first 40 sec? rocket during the first 40 sec? What is the instantaneous velocity of the What is the instantaneous velocity of the

rocket at the 40 sec mark?rocket at the 40 sec mark?


2: (cont)2: (cont)

How high does the rocket travel?How high does the rocket travel?

Knowns:

s = 5t3 ft t = 40 sec

s= 5 (40)3 s = 320, 000 ft


2: (cont)2: (cont)

What is the average velocity of the What is the average velocity of the rocket during the first 40 sec?rocket during the first 40 sec?

V average = f(t1) – f(t0)

t1 - t0

V average = 320000ft – 0

40 sec

V average = 8000 ft/sec


2: (cont)2: (cont)

What is the instantaneous velocity of What is the instantaneous velocity of the rocket at the 40 sec mark?the rocket at the 40 sec mark?

V instantaneous = lim f(t1) – f(t0)

t1 - t0 t1t0

V instantaneous = lim 5*(t1)3 – [5*403]

t1 – 40 t140

V instantaneous = lim 5*(t1)3 – 320000

t1 – 40 t140

V inst = lim 5t2 +200t + 8000 t140

V inst = 24,000 ft/sec

3.23.2The DerivativeThe Derivative

Definition of the derivativeDefinition of the derivativeTangent LinesTangent LinesThe Derivative of fThe Derivative of f

with Respect to xwith Respect to xDifferentiabilityDifferentiabilityDerivative NotationDerivative NotationDerivatives at the Derivatives at the

endpoints of an intervalendpoints of an interval

Definition of the Definition of the DerivativeDerivative

The derivative of f at x = xThe derivative of f at x = x00 is denoted by is denoted by

f ’(xf ’(x00) = lim f(x) = lim f(x11) – f(x) – f(x22))

xx11 – x – x22

x1 x2

Assuming this limit exists,Assuming this limit exists,

f ‘ (xf ‘ (x00) = the slope of f at (x) = the slope of f at (x00, f(x, f(x00))))

Tangent LinesTangent LinesThe tangent line to the graph of f at (xThe tangent line to the graph of f at (x00, ,

f(xf(x00)) is the line whose equation is:)) is the line whose equation is:

y - f(xy - f(x00) = f‘(x) = f‘(x00) * ( x - x) * ( x - x00 ) )

The Derivative of f with The Derivative of f with Respect to xRespect to x

f ’(x) = lim f(w) – f(x)f ’(x) = lim f(w) – f(x)

w – xw – xw w x x

DifferentiabilityDifferentiabilityFor a given function, if xFor a given function, if x00 is not in the domain is not in the domain

of f, or if the limit does not exist at xof f, or if the limit does not exist at x00, than the , than the

function is function is not differentiablenot differentiable at x at x00

The most common instances of The most common instances of nondifferentiability occur at a:nondifferentiability occur at a:NOTE: If f is differeniable at x=xNOTE: If f is differeniable at x=x00, then f , then f

must also be continuous at xmust also be continuous at x00

Derivative NotationDerivative Notation

““the derivative of f(x) the derivative of f(x) with respect to x”with respect to x”

Derivatives at the Derivatives at the Endpoints of an IntervalEndpoints of an Interval

If a function f is defined on a closed If a function f is defined on a closed interval [a, b], then the derivative interval [a, b], then the derivative f’(x) f’(x) is is not defined at the endpoints becausenot defined at the endpoints because

f ’(x) = lim f(w) – f(x)f ’(x) = lim f(w) – f(x)

w – xw – xwx

is a two-sided limit.is a two-sided limit.

Therefore, define the derivatives using Therefore, define the derivatives using one-sided, right and left hand, limitsone-sided, right and left hand, limits

Derivatives at the Derivatives at the Endpoints of an IntervalEndpoints of an IntervalA function f is differentiable on intervals A function f is differentiable on intervals

[a, b][a, b][a, +[a, +∞)∞)(-∞, b] (-∞, b] [a, b)[a, b)(a, b](a, b]

if f is differentiable at all numbers inside if f is differentiable at all numbers inside the interval, and at the endpoints (from the interval, and at the endpoints (from the left or right)the left or right)

Examples!!Examples!!1: Given that f(3) = -1 and f’(x) = 5, find 1: Given that f(3) = -1 and f’(x) = 5, find

an equation for the tangent line to the an equation for the tangent line to the graph of y = f(x) at x=3graph of y = f(x) at x=3

KNOWNS:

F’(x) = slope of the tangent line = 5

Point given = (3, -1)

USING POINT SLOPE FORM:

y + 1 = (5) (x – 3)y = 5x - 16

Examples!!Examples!!2: For f(x)=3x2: For f(x)=3x2 2 , find f’(x), and then find , find f’(x), and then find

the equation of the tangent line to the equation of the tangent line to y=f(x) at x = 3y=f(x) at x = 3

f’(x) = 6a

f’(x) = lim f(x) – f(a)

x - axa

f’(x) = lim 3x2 – 3a2

x - axa

f’(x) = lim 3x + 3a

xa

KNOWNS:

f’(x) = slope of tangent line (6a)

point (3, 27)

POINT SLOPE FORM:

y – 27 = (18) (x – 3)

y = 18x - 27

3.33.3Techniques of DiffereniationTechniques of Differeniation

Basic PropertiesBasic PropertiesThe Power RuleThe Power RuleThe Product RuleThe Product RuleThe Quotient RuleThe Quotient Rule

Basic PropertiesBasic Properties

The Power RuleThe Power Rule

The Product RuleThe Product Rule

The Quotient RuleThe Quotient Rule


1: Find dy/dx of y= (x-3) (x1: Find dy/dx of y= (x-3) (x44 + 7) + 7)

Solve this using the product rule:

Let f(x) = (x-3) and g(x)= (x4 + 7)

y’ = 5x4 - 12x3 + 7


Solve this using the quotient rule:

Let f(x) = 4x + 1 and g(x) = x2 - 5

- (4x2 + 2x + 20)x4 – 10x2 + 25

y’ =

4x + 14x + 1xx22 - 5 - 52: Find dy/dx of y =2: Find dy/dx of y =

3.43.4Derivatives of Derivatives of Trigonometric FunctionsTrigonometric Functions

Derivatives of the Trigonometric Functions Derivatives of the Trigonometric Functions (sinx, cosx, tanx, secx, cotx, cscx)(sinx, cosx, tanx, secx, cotx, cscx)

Derivatives of Derivatives of Trigonometric Functions!Trigonometric Functions!

Examples!!Examples!!sin sin x x sec sec xx1 + 1 + x x tan tan xx

1: Find dy/dx of y =1: Find dy/dx of y =

Solve this using the quotient and product rules:

Simplify.

1(1 + x tanx)2

y’ =


Solve this using the product rule:

2: Find y2: Find y´́ (x) of y = (x) of y = xx33 sin sin x x – 5 cos – 5 cos xx

dy/dx = x3 (cos x) + (sin x)(3x2) + 5 sin x

3.53.5The Chain RuleThe Chain Rule

Derivatives of CompositionsDerivatives of CompositionsThe Chain RuleThe Chain RuleAn Alternate ApproachAn Alternate Approach

Derivatives of Derivatives of CompositionsCompositions

If you know the derivative of f and g, If you know the derivative of f and g, how can you use these to find the how can you use these to find the derivative of the composition of f derivative of the composition of f ° g?° g?

Chain Rule!Chain Rule!

If g is differentiable at x and f is If g is differentiable at x and f is differentiable at g(x), then the differentiable at g(x), then the composition f ° g is differentiable at xcomposition f ° g is differentiable at x

If y = If y = f(g(x)) f(g(x)) and u = and u = g(x) g(x)

then y = f(u)then y = f(u)

An Alternative ApproachAn Alternative ApproachSometimes it is easier to write the chain Sometimes it is easier to write the chain

rule as:rule as:

g(x) is the inside function g(x) is the inside function

““The derivative of The derivative of f(g(x)) f(g(x)) is the derivative is the derivative of the outside function evaluated at the of the outside function evaluated at the inside function times the derivative of the inside function times the derivative of the inside function”inside function”

f(x) is the outside functionf(x) is the outside function

An Alternative ApproachAn Alternative Approach

That is:That is:

An Alternative ApproachAn Alternative Approach

Substituting u = Substituting u = g(x) g(x) you getyou get


1: Find dy/dx of y = (51: Find dy/dx of y = (5xx + + 88))1313((xx33 + 7 + 7xx))1212

Use the chain rule, and product rule

dy/dx = [(5x +8)13][12(x3 + 7x)11(3x2 + 7)] +

[(x3 + 7x)12][13(5x + 8)12(5)]

dy/dx = 12(5x +8)13(x3 + 7x)11(3x2 + 7) +

65(x3 + 7x)12 (5x + 8)12

3.63.6Implicit DifferentiationImplicit Differentiation

Explicit versus ImplicitExplicit versus Implicit Implicit DifferentiationImplicit Differentiation

Explicit Versus ImplicitExplicit Versus Implicit

““A function in the form A function in the form y = f(x) y = f(x) is said to is said to define y define y explicitly explicitly as a function of x as a function of x because the variable y appears alone because the variable y appears alone on one side of the equation.”on one side of the equation.”

““If a function is defined by an equation If a function is defined by an equation in which y is not alone on one side, we in which y is not alone on one side, we say that the function defines y say that the function defines y implicitlyimplicitly””


Implicit:Implicit:yx + y + 1 = xyx + y + 1 = x

NOTE: The implicit function can NOTE: The implicit function can sometimes by rewritten into an sometimes by rewritten into an explicit function explicit function

Explicit:Explicit:y = (x-1) / (x+1)y = (x-1) / (x+1)


““A given equation in x and y defines the A given equation in x and y defines the function f function f implicitlyimplicitly if the graph of if the graph of y = y = f(x) f(x) coincides with a portion of the coincides with a portion of the graph of the equation”graph of the equation”

Explicit Versus ImplicitExplicit Versus ImplicitSo, for example the graph of xSo, for example the graph of x22 + y + y22 = 1 = 1

defines the functions defines the functions

ff11(x) = (x) = √(1-x√(1-x22))

ff22(x) = -√(1-x(x) = -√(1-x22) )

implicitly, since the graphs of these implicitly, since the graphs of these functions are contained in the circle functions are contained in the circle xx22 + y + y22 = 1 = 1


Implicit DifferentiationImplicit Differentiation

Usually, it is not necessary to solve an Usually, it is not necessary to solve an equation for equation for y y in terms of in terms of xx in order to in order to differentiate the functions defined differentiate the functions defined implicitly by the equationimplicitly by the equation


1: Find dy/dx for sin(x1: Find dy/dx for sin(x22yy22) = x) = x

cos(x2y2) [x2(2y) + y2 (2x)] = 1dydx

dydx cos(x2y2)

2yx2

= 1 - 2xy2

= 1- 2xy2cos(x2y2)

2yx2cos(x2y2)

Examples!!Examples!!2: Find d2: Find d22y/dxy/dx22 for x for x33yy33 – 4 = 0 – 4 = 0

x3(3y2) + y3(3x2) = 0dxdy

-y3x2

x3y2dxdy =

x -y

=

(x)(-1 ) - (-y)(1)

dx2d2y =

x2

dxdy

x2

x -y

dx2d2y

=-x( ) + y

dx2d2y

= 2y

x2

3.73.7Related RatesRelated RatesDifferentiating Equations to Relate RatesDifferentiating Equations to Relate Rates

Differentiating Equations Differentiating Equations to Relate Ratesto Relate Rates

Strategy for Solving Related RatesStrategy for Solving Related Rates

Step 1: “Identify the rates of change that Step 1: “Identify the rates of change that are known and the rate of change that are known and the rate of change that is to be found. Interpret each rate as a is to be found. Interpret each rate as a derivative of a variable with respect to derivative of a variable with respect to time, and provide a description of time, and provide a description of each variable involved”.each variable involved”.



Step 2: “Find an equation relating those Step 2: “Find an equation relating those quantities whose rates are identified quantities whose rates are identified in Step 1. In a geometric problem, this in Step 1. In a geometric problem, this is aided by drawing an appropriately is aided by drawing an appropriately labeled figure that illustrates a labeled figure that illustrates a relationship involving these relationship involving these quantities”.quantities”.



Step 3: “Obtain an equation involving Step 3: “Obtain an equation involving the rates in Step 1 by differentiating the rates in Step 1 by differentiating both sides of the equation in Step 2 both sides of the equation in Step 2 with respect to the time variable”.with respect to the time variable”.



Step 4: “Evaluate the equation found in Step 4: “Evaluate the equation found in Step 3 using the known values for the Step 3 using the known values for the quantities and their rates of change at quantities and their rates of change at the moment in question”.the moment in question”.



Step 5: “Solve for the value of the Step 5: “Solve for the value of the remaining rate of change at this remaining rate of change at this moment”.moment”.

Example!!Example!!1: Sand pouring from a chute forms a 1: Sand pouring from a chute forms a

conical pile whose height is always equal conical pile whose height is always equal to the diameter. If the height increases at to the diameter. If the height increases at a constant rate of 5ft/ min, at what rate is a constant rate of 5ft/ min, at what rate is sand pouring from the chute when the sand pouring from the chute when the pile is 10 ft high?pile is 10 ft high?

Example!!Example!!

STEP1.

dtdh

= 5 ft/ min

= ? dV

dth = 10 ft

t = time

h = height of conical pile at a given time

V = amount of sand in conical pile at a given time

1: (cont.)1: (cont.)

Example!!Example!!

STEP2.

V = π h3

12


Example!!Example!!

STEP3.V = π h3

12


STEP4.

dtdV = π h2

4 dtdh

dtdV = π (10ft)2

4(5ft/min) = 125π ft3 / min

STEP5.

Example!!Example!!

2: A 13-ft ladder is 2: A 13-ft ladder is leaning against a wall. leaning against a wall. If the top of the ladder If the top of the ladder slips down the wall at a slips down the wall at a rate of 2 ft/sec, how fast rate of 2 ft/sec, how fast will the foot of the will the foot of the ladder be moving away ladder be moving away from the wall when the from the wall when the top is 5 ft above the top is 5 ft above the ground?ground?

Example!!Example!!

STEP1.

dtdh

= -2 ft/ sec

= ? dD

dth = 5 ft

t = time

h = height of the top of the ladder against the wall

D = distance of the foot of the ladder from the base of the wall


Example!!Example!!

STEP2.


D2 + h2 = 132

D2 + h2 = 169

Example!!Example!!

STEP3.


STEP4. STEP5.

D2 + h2 = 169

= 5/6 ft / secdtdD

2D + 2h = 0dtdD

dtdh

24ft + 10ft (-2 ft/sec) = 0dtdD

(note: at h=5, D = 12)

3.83.8Local Linear Approximation; Local Linear Approximation; DifferentialsDifferentials

Local Linear ApproximationLocal Linear ApproximationDifferentialsDifferentials

Local Linear Local Linear ApproximationApproximation

““Linear Approximation may be Linear Approximation may be described informally in terms of the described informally in terms of the behavior of the graph of behavior of the graph of f f under under magnification: if magnification: if f f is differentiable at is differentiable at xx00, then stronger and stronger , then stronger and stronger magnifications at a point, P, magnifications at a point, P, eventually make the curve segment eventually make the curve segment containing P look more and more like containing P look more and more like a nonvertical line segment, that line a nonvertical line segment, that line being the tangent line to the graph of being the tangent line to the graph of ff at P.” at P.”


A function that is differentiable at xA function that is differentiable at x00

is said to be is said to be locally linear locally linear at the at the point P (xpoint P (x00, f(x, f(x00))))

As you zoom As you zoom closer to a closer to a point P, the point P, the function looks function looks more and more and more linearmore linear


Assume that a function Assume that a function f f is is differentiable at xdifferentiable at x00, and remember , and remember

that the equation of the tangent line that the equation of the tangent line at the point P (xat the point P (x00, f(x, f(x00)) is:)) is:


Since the tangent line closely Since the tangent line closely approximates the graph of approximates the graph of f f for for values of values of x x near xnear x00, that means that , that means that

provided provided x x is near xis near x00, then:, then:

This is called the This is called the local linear local linear approximation approximation of f at xof f at x00


By rewriting the formulaBy rewriting the formula

with with ∆x = x -∆x = x - x x00, you get: , you get:


Generally, the Generally, the accuracy of the accuracy of the local linear local linear approximation approximation to f(x) at xto f(x) at x00 will will

deteriorate as c deteriorate as c gets gets progressively progressively farther from xfarther from x00..

DifferentialsDifferentialsEarly in the development of calculus, the Early in the development of calculus, the

symbols “symbols “dydy” and “” and “dxdx” represented ” represented “infinitely small changes” in the variables “infinitely small changes” in the variables y y and and xx. The derivative . The derivative dy/dxdy/dx was thought was thought to be a ratio of these infinitely small to be a ratio of these infinitely small changes. However, the precise meaning changes. However, the precise meaning is logically elusive.is logically elusive.

Our goal is to define the symbols Our goal is to define the symbols dy dy and and dxdx so that so that dy/dx dy/dx can actually be treated as a can actually be treated as a ratioratio

DifferentialsDifferentialsThe variable The variable dxdx is called the is called the differential of xdifferential of x. .

If we are given a function y = If we are given a function y = f(x)f(x) tha is tha is differentiable at x = xdifferentiable at x = x00, then we define the , then we define the

differential of f at xdifferential of f at x00 to be the function of to be the function of

dx dx given by the formula:given by the formula:

DifferentialsDifferentials

The symbol The symbol dydy is simply the dependant is simply the dependant variable of this function, and is called variable of this function, and is called the differential of y. the differential of y. dydy is proportional is proportional to to dx dx with constant of proportionality f ‘ with constant of proportionality f ‘ (x(x00). If ). If dx dx is not 0, you can obtainis not 0, you can obtain

DifferentialsDifferentials

Because Because f ‘ (x)f ‘ (x) is equal to the slope of the is equal to the slope of the tangent line to the graph of tangent line to the graph of ff at the point at the point ((xx,,ff((xx)), the differentials )), the differentials dy dy and and dx dx can can be described as the rise and run of this be described as the rise and run of this tangent line.tangent line.

EXAMPLES!!EXAMPLES!!

1: Find the local linear approximation of 1: Find the local linear approximation of xx33 at x at x00 = 1 = 1

y ≈ 1 + 3(x-1) y ≈ 3x + 2

EXAMPLES!!EXAMPLES!!2: Use an appropriate local linear 2: Use an appropriate local linear

approximation to estimate the value of approximation to estimate the value of √ (36.03)√ (36.03)

f(36.03) ≈ f(36) + (1/12)(36.03-36)

f(36.06) ≈ 6.0025

Let f(x)=√(x) f ‘ (x)= (1/2)x-(1/2)

f(36.03) ≈ 6 + (1/12)(.03)

EXAMPLES!!EXAMPLES!!

3: Find the differential 3: Find the differential dy dy for y = xcosx for y = xcosx

dy/dx = -xsinx + cosx

dy = (-xsinx + cosx) dx

BIBLIOGRAPHY!!!BIBLIOGRAPHY!!!

http://mathcs.holycross.edu/~spl/old_courses/131_fall_2005/http://mathcs.holycross.edu/~spl/old_courses/131_fall_2005/tangent_line.giftangent_line.gif

http://www.coolschool.ca/lor/CALC12/unit3/U03L08/example_07.gifhttp://www.coolschool.ca/lor/CALC12/unit3/U03L08/example_07.gif http://www.clas.ucsb.edu/staff/Lee/Tangent%20and%20Derivative.gifhttp://www.clas.ucsb.edu/staff/Lee/Tangent%20and%20Derivative.gif http://images.search.yahoo.com/search/images/view?back=http%3Ahttp://images.search.yahoo.com/search/images/view?back=http%3A

%2F%2Fimages.search.yahoo.com%2Fsearch%2Fimages%3Fp%2F%2Fimages.search.yahoo.com%2Fsearch%2Fimages%3Fp%3Dproduct%2Brule%2Bcalculus%26ei%3DUTF-8%26fr%3Dyfp-t-%3Dproduct%2Brule%2Bcalculus%26ei%3DUTF-8%26fr%3Dyfp-t-501%26x%3Dwrt&w=309&h=272&imgurl=www.karlscalculus.org501%26x%3Dwrt&w=309&h=272&imgurl=www.karlscalculus.org%2Fqrule_still.gif&rurl=http%3A%2F%2Fwww.karlscalculus.org%2Fqrule_still.gif&rurl=http%3A%2F%2Fwww.karlscalculus.org%2Fcalc4_3.html&size=15.1kB&name=qrule_still.gif&p=product+rule+%2Fcalc4_3.html&size=15.1kB&name=qrule_still.gif&p=product+rule+calculus&type=gif&no=4&tt=53&oid=0077160168cbaf58&ei=UTF-8calculus&type=gif&no=4&tt=53&oid=0077160168cbaf58&ei=UTF-8

Chapter 3 The Derivative By: Kristen Whaley. 3.1 Slopes and Rates of Change Average Velocity Instantaneous Velocity Average Rate of Change Instantaneous.

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