83
. . . . . . Section 3.3 Derivatives of Exponential and Logarithmic Functions V63.0121.006/016, Calculus I March 2, 2010 Announcements I Review sessions: tonight, 7:30 in CIWW 202 and 517; tomorrow, 7:00 in CIWW 109 I Midterm is March 4, covering §§1.1–2.5 I Recitation this week will cover §§3.1–3.2

Lesson 13: Derivatives of Logarithmic and Exponential Functions

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Page 1: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Section3.3DerivativesofExponentialand

LogarithmicFunctions

V63.0121.006/016, CalculusI

March2, 2010

Announcements

I Reviewsessions: tonight, 7:30inCIWW 202and517;tomorrow, 7:00inCIWW 109

I MidtermisMarch4, covering§§1.1–2.5I Recitationthisweekwillcover§§3.1–3.2

Page 2: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Announcements

I Reviewsessions: tonight, 7:30inCIWW 202and517;tomorrow, 7:00inCIWW 109

I MidtermisMarch4, covering§§1.1–2.5I Recitationthisweekwillcover§§3.1–3.2

Page 3: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Outline

“Recall”Section3.1–3.2

DerivativeofthenaturalexponentialfunctionExponentialGrowth

Derivativeofthenaturallogarithmfunction

DerivativesofotherexponentialsandlogarithmsOtherexponentialsOtherlogarithms

LogarithmicDifferentiationThepowerruleforirrationalpowers

Page 4: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Conventionsonexponentialfunctions

Let a beapositiverealnumber.I If n isapositivewholenumber, then an = a · a · · · · · a︸ ︷︷ ︸

n factors

I a0 = 1.

I Foranyrealnumber r, a−r =1ar.

I Foranypositivewholenumber n, a1/n = n√a.

Thereisonlyonecontinuousfunctionwhichsatisfiesalloftheabove. Wecallitthe exponentialfunction withbase a.

Page 5: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Propertiesofexponentialfunctions

TheoremIf a > 0 and a ̸= 1, then f(x) = ax isacontinuousfunctionwithdomain R andrange (0,∞). Inparticular, ax > 0 forall x. Ifa,b > 0 and x, y ∈ R, then

I ax+y = axay

I ax−y =ax

ayI (ax)y = axy

I (ab)x = axbx

Page 6: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsofvariousexponentialfunctions

. .x

.y

.y = 1x

.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x

Page 7: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsofvariousexponentialfunctions

. .x

.y

.y = 1x

.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x

Page 8: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsofvariousexponentialfunctions

. .x

.y

.y = 1x

.y = 2x

.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x

Page 9: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsofvariousexponentialfunctions

. .x

.y

.y = 1x

.y = 2x.y = 3x

.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x

Page 10: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsofvariousexponentialfunctions

. .x

.y

.y = 1x

.y = 2x.y = 3x.y = 10x

.y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x

Page 11: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsofvariousexponentialfunctions

. .x

.y

.y = 1x

.y = 2x.y = 3x.y = 10x .y = 1.5x

.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x

Page 12: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsofvariousexponentialfunctions

. .x

.y

.y = 1x

.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x

.y = (1/3)x .y = (1/10)x.y = (2/3)x

Page 13: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsofvariousexponentialfunctions

. .x

.y

.y = 1x

.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x

.y = (1/10)x.y = (2/3)x

Page 14: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsofvariousexponentialfunctions

. .x

.y

.y = 1x

.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x

.y = (2/3)x

Page 15: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsofvariousexponentialfunctions

. .x

.y

.y = 1x

.y = 2x.y = 3x.y = 10x .y = 1.5x.y = (1/2)x.y = (1/3)x .y = (1/10)x.y = (2/3)x

Page 16: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Themagicnumber

Definition

e = limn→∞

(1+

1n

)n

= limh→0+

(1+ h)1/h

Page 17: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Existenceof eSeeAppendixB

I Wecanexperimentallyverifythatthisnumberexistsandis

e ≈ 2.718281828459045 . . .

I e isirrationalI e is transcendental

n(1+

1n

)n

1 22 2.25

3 2.3703710 2.59374100 2.704811000 2.71692106 2.71828

Page 18: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Existenceof eSeeAppendixB

I Wecanexperimentallyverifythatthisnumberexistsandis

e ≈ 2.718281828459045 . . .

I e isirrationalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.37037

10 2.59374100 2.704811000 2.71692106 2.71828

Page 19: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Existenceof eSeeAppendixB

I Wecanexperimentallyverifythatthisnumberexistsandis

e ≈ 2.718281828459045 . . .

I e isirrationalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374

100 2.704811000 2.71692106 2.71828

Page 20: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Existenceof eSeeAppendixB

I Wecanexperimentallyverifythatthisnumberexistsandis

e ≈ 2.718281828459045 . . .

I e isirrationalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.70481

1000 2.71692106 2.71828

Page 21: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Existenceof eSeeAppendixB

I Wecanexperimentallyverifythatthisnumberexistsandis

e ≈ 2.718281828459045 . . .

I e isirrationalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692

106 2.71828

Page 22: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Existenceof eSeeAppendixB

I Wecanexperimentallyverifythatthisnumberexistsandis

e ≈ 2.718281828459045 . . .

I e isirrationalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 2.71828

Page 23: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Existenceof eSeeAppendixB

I Wecanexperimentallyverifythatthisnumberexistsandis

e ≈ 2.718281828459045 . . .

I e isirrationalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 2.71828

Page 24: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Existenceof eSeeAppendixB

I Wecanexperimentallyverifythatthisnumberexistsandis

e ≈ 2.718281828459045 . . .

I e isirrational

I e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 2.71828

Page 25: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Existenceof eSeeAppendixB

I Wecanexperimentallyverifythatthisnumberexistsandis

e ≈ 2.718281828459045 . . .

I e isirrationalI e is transcendental

n(1+

1n

)n

1 22 2.253 2.3703710 2.59374100 2.704811000 2.71692106 2.71828

Page 26: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Logarithms

Definition

I Thebase a logarithm loga x istheinverseofthefunction ax

y = loga x ⇐⇒ x = ay

I Thenaturallogarithm ln x istheinverseof ex. Soy = ln x ⇐⇒ x = ey.

Facts

(i) loga(x · x′) = loga x+ loga x

(ii) loga( xx′

)= loga x− loga x

(iii) loga(xr) = r loga x

Page 27: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Logarithms

Definition

I Thebase a logarithm loga x istheinverseofthefunction ax

y = loga x ⇐⇒ x = ay

I Thenaturallogarithm ln x istheinverseof ex. Soy = ln x ⇐⇒ x = ey.

Facts

(i) loga(x · x′) = loga x+ loga x

(ii) loga( xx′

)= loga x− loga x

(iii) loga(xr) = r loga x

Page 28: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Logarithms

Definition

I Thebase a logarithm loga x istheinverseofthefunction ax

y = loga x ⇐⇒ x = ay

I Thenaturallogarithm ln x istheinverseof ex. Soy = ln x ⇐⇒ x = ey.

Facts

(i) loga(x · x′) = loga x+ loga x

(ii) loga( xx′

)= loga x− loga x

(iii) loga(xr) = r loga x

Page 29: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Logarithms

Definition

I Thebase a logarithm loga x istheinverseofthefunction ax

y = loga x ⇐⇒ x = ay

I Thenaturallogarithm ln x istheinverseof ex. Soy = ln x ⇐⇒ x = ey.

Facts

(i) loga(x · x′) = loga x+ loga x

(ii) loga( xx′

)= loga x− loga x

(iii) loga(xr) = r loga x

Page 30: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Logarithmsconvertproductstosums

I Suppose y = loga x and y′ = loga x′

I Then x = ay and x′ = ay′

I So xx′ = ayay′= ay+y′

I Therefore

loga(xx′) = y+ y′ = loga x+ loga x

Page 31: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsoflogarithmicfunctions

. .x

.y.y = 2x

.y = log2 x

. .(0, 1)

..(1, 0)

.y = 3x

.y = log3 x

.y = 10x

.y = log10 x

.y = ex

.y = ln x

Page 32: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsoflogarithmicfunctions

. .x

.y.y = 2x

.y = log2 x

. .(0, 1)

..(1, 0)

.y = 3x

.y = log3 x

.y = 10x

.y = log10 x

.y = ex

.y = ln x

Page 33: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsoflogarithmicfunctions

. .x

.y.y = 2x

.y = log2 x

. .(0, 1)

..(1, 0)

.y = 3x

.y = log3 x

.y = 10x

.y = log10 x

.y = ex

.y = ln x

Page 34: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Graphsoflogarithmicfunctions

. .x

.y.y = 2x

.y = log2 x

. .(0, 1)

..(1, 0)

.y = 3x

.y = log3 x

.y = 10x

.y = log10 x

.y = ex

.y = ln x

Page 35: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Changeofbaseformulaforexponentials

FactIf a > 0 and a ̸= 1, then

loga x =ln xln a

Proof.

I If y = loga x, then x = ay

I So ln x = ln(ay) = y ln aI Therefore

y = loga x =ln xln a

Page 36: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Changeofbaseformulaforexponentials

FactIf a > 0 and a ̸= 1, then

loga x =ln xln a

Proof.

I If y = loga x, then x = ay

I So ln x = ln(ay) = y ln aI Therefore

y = loga x =ln xln a

Page 37: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Outline

“Recall”Section3.1–3.2

DerivativeofthenaturalexponentialfunctionExponentialGrowth

Derivativeofthenaturallogarithmfunction

DerivativesofotherexponentialsandlogarithmsOtherexponentialsOtherlogarithms

LogarithmicDifferentiationThepowerruleforirrationalpowers

Page 38: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

DerivativesofExponentialFunctions

FactIf f(x) = ax, then f′(x) = f′(0)ax.

Proof.Followyournose:

f′(x) = limh→0

f(x+ h)− f(x)h

= limh→0

ax+h − ax

h

= limh→0

axah − ax

h= ax · lim

h→0

ah − 1h

= ax · f′(0).

Toreiterate: thederivativeofanexponentialfunctionisaconstant times thatfunction. Muchdifferentfrompolynomials!

Page 39: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

DerivativesofExponentialFunctions

FactIf f(x) = ax, then f′(x) = f′(0)ax.

Proof.Followyournose:

f′(x) = limh→0

f(x+ h)− f(x)h

= limh→0

ax+h − ax

h

= limh→0

axah − ax

h= ax · lim

h→0

ah − 1h

= ax · f′(0).

Toreiterate: thederivativeofanexponentialfunctionisaconstant times thatfunction. Muchdifferentfrompolynomials!

Page 40: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

DerivativesofExponentialFunctions

FactIf f(x) = ax, then f′(x) = f′(0)ax.

Proof.Followyournose:

f′(x) = limh→0

f(x+ h)− f(x)h

= limh→0

ax+h − ax

h

= limh→0

axah − ax

h= ax · lim

h→0

ah − 1h

= ax · f′(0).

Toreiterate: thederivativeofanexponentialfunctionisaconstant times thatfunction. Muchdifferentfrompolynomials!

Page 41: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Thefunnylimitinthecaseof eRememberthedefinitionof e:

e = limn→∞

(1+

1n

)n

= limh→0

(1+ h)1/h

Question

Whatis limh→0

eh − 1h

?

AnswerIf h issmallenough, e ≈ (1+ h)1/h. So

eh − 1h

≈[(1+ h)1/h

]h − 1h

=(1+ h)− 1

h=

hh= 1

Sointhelimitwegetequality: limh→0

eh − 1h

= 1

Page 42: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Thefunnylimitinthecaseof eRememberthedefinitionof e:

e = limn→∞

(1+

1n

)n

= limh→0

(1+ h)1/h

Question

Whatis limh→0

eh − 1h

?

AnswerIf h issmallenough, e ≈ (1+ h)1/h. So

eh − 1h

≈[(1+ h)1/h

]h − 1h

=(1+ h)− 1

h=

hh= 1

Sointhelimitwegetequality: limh→0

eh − 1h

= 1

Page 43: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Thefunnylimitinthecaseof eRememberthedefinitionof e:

e = limn→∞

(1+

1n

)n

= limh→0

(1+ h)1/h

Question

Whatis limh→0

eh − 1h

?

AnswerIf h issmallenough, e ≈ (1+ h)1/h. So

eh − 1h

≈[(1+ h)1/h

]h − 1h

=(1+ h)− 1

h=

hh= 1

Sointhelimitwegetequality: limh→0

eh − 1h

= 1

Page 44: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Derivativeofthenaturalexponentialfunction

From

ddx

ax =

(limh→0

ah − 1h

)ax and lim

h→0

eh − 1h

= 1

weget:

Theorem

ddx

ex = ex

Page 45: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

ExponentialGrowth

I Commonlymisusedtermtosaysomethinggrowsexponentially

I Itmeanstherateofchange(derivative)isproportionaltothecurrentvalue

I Examples: Naturalpopulationgrowth, compoundedinterest,socialnetworks

Page 46: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Examples

ExamplesFindthesederivatives:

I e3x

I ex2

I x2ex

Solution

Iddx

e3x = 3e3x

Iddx

ex2= ex

2 ddx

(x2) = 2xex2

Iddx

x2ex = 2xex + x2ex

Page 47: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Examples

ExamplesFindthesederivatives:

I e3x

I ex2

I x2ex

Solution

Iddx

e3x = 3e3x

Iddx

ex2= ex

2 ddx

(x2) = 2xex2

Iddx

x2ex = 2xex + x2ex

Page 48: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Examples

ExamplesFindthesederivatives:

I e3x

I ex2

I x2ex

Solution

Iddx

e3x = 3e3x

Iddx

ex2= ex

2 ddx

(x2) = 2xex2

Iddx

x2ex = 2xex + x2ex

Page 49: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Examples

ExamplesFindthesederivatives:

I e3x

I ex2

I x2ex

Solution

Iddx

e3x = 3e3x

Iddx

ex2= ex

2 ddx

(x2) = 2xex2

Iddx

x2ex = 2xex + x2ex

Page 50: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Outline

“Recall”Section3.1–3.2

DerivativeofthenaturalexponentialfunctionExponentialGrowth

Derivativeofthenaturallogarithmfunction

DerivativesofotherexponentialsandlogarithmsOtherexponentialsOtherlogarithms

LogarithmicDifferentiationThepowerruleforirrationalpowers

Page 51: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Derivativeofthenaturallogarithmfunction

Let y = ln x. Thenx = ey so

eydydx

= 1

=⇒ dydx

=1ey

=1x

So:

Fact

ddx

ln x =1x

. .x

.y

.ln x

.1x

Page 52: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Derivativeofthenaturallogarithmfunction

Let y = ln x. Thenx = ey so

eydydx

= 1

=⇒ dydx

=1ey

=1x

So:

Fact

ddx

ln x =1x

. .x

.y

.ln x

.1x

Page 53: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Derivativeofthenaturallogarithmfunction

Let y = ln x. Thenx = ey so

eydydx

= 1

=⇒ dydx

=1ey

=1x

So:

Fact

ddx

ln x =1x

. .x

.y

.ln x

.1x

Page 54: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Derivativeofthenaturallogarithmfunction

Let y = ln x. Thenx = ey so

eydydx

= 1

=⇒ dydx

=1ey

=1x

So:

Fact

ddx

ln x =1x

. .x

.y

.ln x

.1x

Page 55: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Derivativeofthenaturallogarithmfunction

Let y = ln x. Thenx = ey so

eydydx

= 1

=⇒ dydx

=1ey

=1x

So:

Fact

ddx

ln x =1x

. .x

.y

.ln x

.1x

Page 56: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Derivativeofthenaturallogarithmfunction

Let y = ln x. Thenx = ey so

eydydx

= 1

=⇒ dydx

=1ey

=1x

So:

Fact

ddx

ln x =1x

. .x

.y

.ln x

.1x

Page 57: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

TheTowerofPowers

y y′

x3 3x2

x2 2x1

x1 1x0

x0 0

? ?

x−1 −1x−2

x−2 −2x−3

I Thederivativeofapowerfunctionisapowerfunctionofonelowerpower

I Eachpowerfunctionisthederivativeofanotherpowerfunction, exceptx−1

I ln x fillsinthisgapprecisely.

Page 58: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

TheTowerofPowers

y y′

x3 3x2

x2 2x1

x1 1x0

x0 0

? x−1

x−1 −1x−2

x−2 −2x−3

I Thederivativeofapowerfunctionisapowerfunctionofonelowerpower

I Eachpowerfunctionisthederivativeofanotherpowerfunction, exceptx−1

I ln x fillsinthisgapprecisely.

Page 59: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

TheTowerofPowers

y y′

x3 3x2

x2 2x1

x1 1x0

x0 0

ln x x−1

x−1 −1x−2

x−2 −2x−3

I Thederivativeofapowerfunctionisapowerfunctionofonelowerpower

I Eachpowerfunctionisthederivativeofanotherpowerfunction, exceptx−1

I ln x fillsinthisgapprecisely.

Page 60: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Outline

“Recall”Section3.1–3.2

DerivativeofthenaturalexponentialfunctionExponentialGrowth

Derivativeofthenaturallogarithmfunction

DerivativesofotherexponentialsandlogarithmsOtherexponentialsOtherlogarithms

LogarithmicDifferentiationThepowerruleforirrationalpowers

Page 61: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Otherlogarithms

Example

Useimplicitdifferentiationtofindddx

ax.

SolutionLet y = ax, so

ln y = ln ax = x ln a

Differentiateimplicitly:

1ydydx

= ln a =⇒ dydx

= (ln a)y = (ln a)ax

Beforeweshowed y′ = y′(0)y, sonowweknowthat

ln 2 = limh→0

2h − 1h

≈ 0.693 ln 3 = limh→0

3h − 1h

≈ 1.10

Page 62: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Otherlogarithms

Example

Useimplicitdifferentiationtofindddx

ax.

SolutionLet y = ax, so

ln y = ln ax = x ln a

Differentiateimplicitly:

1ydydx

= ln a =⇒ dydx

= (ln a)y = (ln a)ax

Beforeweshowed y′ = y′(0)y, sonowweknowthat

ln 2 = limh→0

2h − 1h

≈ 0.693 ln 3 = limh→0

3h − 1h

≈ 1.10

Page 63: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Otherlogarithms

Example

Useimplicitdifferentiationtofindddx

ax.

SolutionLet y = ax, so

ln y = ln ax = x ln a

Differentiateimplicitly:

1ydydx

= ln a =⇒ dydx

= (ln a)y = (ln a)ax

Beforeweshowed y′ = y′(0)y, sonowweknowthat

ln 2 = limh→0

2h − 1h

≈ 0.693 ln 3 = limh→0

3h − 1h

≈ 1.10

Page 64: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Otherlogarithms

Example

Useimplicitdifferentiationtofindddx

ax.

SolutionLet y = ax, so

ln y = ln ax = x ln a

Differentiateimplicitly:

1ydydx

= ln a =⇒ dydx

= (ln a)y = (ln a)ax

Beforeweshowed y′ = y′(0)y, sonowweknowthat

ln 2 = limh→0

2h − 1h

≈ 0.693 ln 3 = limh→0

3h − 1h

≈ 1.10

Page 65: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Otherlogarithms

Example

Findddx

loga x.

SolutionLet y = loga x, so ay = x. Nowdifferentiateimplicitly:

(ln a)aydydx

= 1 =⇒ dydx

=1

ay ln a=

1x ln a

Anotherwaytoseethisistotakethenaturallogarithm:

ay = x =⇒ y ln a = ln x =⇒ y =ln xln a

Sodydx

=1ln a

1x.

Page 66: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Otherlogarithms

Example

Findddx

loga x.

SolutionLet y = loga x, so ay = x.

Nowdifferentiateimplicitly:

(ln a)aydydx

= 1 =⇒ dydx

=1

ay ln a=

1x ln a

Anotherwaytoseethisistotakethenaturallogarithm:

ay = x =⇒ y ln a = ln x =⇒ y =ln xln a

Sodydx

=1ln a

1x.

Page 67: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Otherlogarithms

Example

Findddx

loga x.

SolutionLet y = loga x, so ay = x. Nowdifferentiateimplicitly:

(ln a)aydydx

= 1 =⇒ dydx

=1

ay ln a=

1x ln a

Anotherwaytoseethisistotakethenaturallogarithm:

ay = x =⇒ y ln a = ln x =⇒ y =ln xln a

Sodydx

=1ln a

1x.

Page 68: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Otherlogarithms

Example

Findddx

loga x.

SolutionLet y = loga x, so ay = x. Nowdifferentiateimplicitly:

(ln a)aydydx

= 1 =⇒ dydx

=1

ay ln a=

1x ln a

Anotherwaytoseethisistotakethenaturallogarithm:

ay = x =⇒ y ln a = ln x =⇒ y =ln xln a

Sodydx

=1ln a

1x.

Page 69: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Moreexamples

Example

Findddx

log2(x2 + 1)

Answer

dydx

=1ln 2

1x2 + 1

(2x) =2x

(ln 2)(x2 + 1)

Page 70: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Moreexamples

Example

Findddx

log2(x2 + 1)

Answer

dydx

=1ln 2

1x2 + 1

(2x) =2x

(ln 2)(x2 + 1)

Page 71: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Outline

“Recall”Section3.1–3.2

DerivativeofthenaturalexponentialfunctionExponentialGrowth

Derivativeofthenaturallogarithmfunction

DerivativesofotherexponentialsandlogarithmsOtherexponentialsOtherlogarithms

LogarithmicDifferentiationThepowerruleforirrationalpowers

Page 72: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

A nastyderivative

Example

Let y =(x2 + 1)

√x+ 3

x− 1. Find y′.

SolutionWeusethequotientrule, andtheproductruleinthenumerator:

y′ =(x− 1)

[2x

√x+ 3+ (x2 + 1)12(x+ 3)−1/2

]− (x2 + 1)

√x+ 3(1)

(x− 1)2

=2x

√x+ 3

(x− 1)+

(x2 + 1)2√x+ 3(x− 1)

− (x2 + 1)√x+ 3

(x− 1)2

Page 73: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

A nastyderivative

Example

Let y =(x2 + 1)

√x+ 3

x− 1. Find y′.

SolutionWeusethequotientrule, andtheproductruleinthenumerator:

y′ =(x− 1)

[2x

√x+ 3+ (x2 + 1)12(x+ 3)−1/2

]− (x2 + 1)

√x+ 3(1)

(x− 1)2

=2x

√x+ 3

(x− 1)+

(x2 + 1)2√x+ 3(x− 1)

− (x2 + 1)√x+ 3

(x− 1)2

Page 74: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Anotherway

y =(x2 + 1)

√x+ 3

x− 1

ln y = ln(x2 + 1) +12ln(x+ 3)− ln(x− 1)

1ydydx

=2x

x2 + 1+

12(x+ 3)

− 1x− 1

So

dydx

=

(2x

x2 + 1+

12(x+ 3)

− 1x− 1

)y

=

(2x

x2 + 1+

12(x+ 3)

− 1x− 1

)(x2 + 1)

√x+ 3

x− 1

Page 75: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Compareandcontrast

I Usingtheproduct, quotient, andpowerrules:

y′ =2x

√x+ 3

(x− 1)+

(x2 + 1)2√x+ 3(x− 1)

− (x2 + 1)√x+ 3

(x− 1)2

I Usinglogarithmicdifferentiation:

y′ =(

2xx2 + 1

+1

2(x+ 3)− 1

x− 1

)(x2 + 1)

√x+ 3

x− 1

I Arethesethesame?I Whichdoyoulikebetter?I Whatkindsofexpressionsarewell-suitedforlogarithmicdifferentiation?

Page 76: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Compareandcontrast

I Usingtheproduct, quotient, andpowerrules:

y′ =2x

√x+ 3

(x− 1)+

(x2 + 1)2√x+ 3(x− 1)

− (x2 + 1)√x+ 3

(x− 1)2

I Usinglogarithmicdifferentiation:

y′ =(

2xx2 + 1

+1

2(x+ 3)− 1

x− 1

)(x2 + 1)

√x+ 3

x− 1

I Arethesethesame?

I Whichdoyoulikebetter?I Whatkindsofexpressionsarewell-suitedforlogarithmicdifferentiation?

Page 77: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Compareandcontrast

I Usingtheproduct, quotient, andpowerrules:

y′ =2x

√x+ 3

(x− 1)+

(x2 + 1)2√x+ 3(x− 1)

− (x2 + 1)√x+ 3

(x− 1)2

I Usinglogarithmicdifferentiation:

y′ =(

2xx2 + 1

+1

2(x+ 3)− 1

x− 1

)(x2 + 1)

√x+ 3

x− 1

I Arethesethesame?I Whichdoyoulikebetter?

I Whatkindsofexpressionsarewell-suitedforlogarithmicdifferentiation?

Page 78: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Compareandcontrast

I Usingtheproduct, quotient, andpowerrules:

y′ =2x

√x+ 3

(x− 1)+

(x2 + 1)2√x+ 3(x− 1)

− (x2 + 1)√x+ 3

(x− 1)2

I Usinglogarithmicdifferentiation:

y′ =(

2xx2 + 1

+1

2(x+ 3)− 1

x− 1

)(x2 + 1)

√x+ 3

x− 1

I Arethesethesame?I Whichdoyoulikebetter?I Whatkindsofexpressionsarewell-suitedforlogarithmicdifferentiation?

Page 79: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Derivativesofpowers

Let y = xx. Whichoftheseistrue?

(A) Since y isapowerfunction, y′ = x · xx−1 = xx.

(B) Since y isanexponentialfunction, y′ = (ln x) · xx

(C) Neither

Page 80: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Derivativesofpowers

Let y = xx. Whichoftheseistrue?

(A) Since y isapowerfunction, y′ = x · xx−1 = xx.

(B) Since y isanexponentialfunction, y′ = (ln x) · xx

(C) Neither

Page 81: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

It’sneither! Orboth?

If y = xx, then

ln y = x ln x

1ydydx

= x · 1x+ ln x = 1+ ln x

dydx

= xx + (ln x)xx

Eachofthesetermsisoneofthewronganswers!

Page 82: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Derivativeofarbitrarypowers

Fact(Thepowerrule)Let y = xr. Then y′ = rxr−1.

Proof.

y = xr =⇒ ln y = r ln x

Nowdifferentiate:

1ydydx

=rx

=⇒ dydx

= ryx= rxr−1

Page 83: Lesson 13: Derivatives of Logarithmic and Exponential Functions

. . . . . .

Derivativeofarbitrarypowers

Fact(Thepowerrule)Let y = xr. Then y′ = rxr−1.

Proof.

y = xr =⇒ ln y = r ln x

Nowdifferentiate:

1ydydx

=rx

=⇒ dydx

= ryx= rxr−1