AP CALCULUS ABChapter 3:DerivativesSection 3.9:

Derivatives of Exponential and Logarithmic Functions

What you’ll learn about Derivative of ex

Derivative of ax

Derivative of ln x Derivative of logax Power Rule for Arbitrary Real Powers

… and why

The relationship between exponential and logarithmic functions provides a powerful differentiation tool called logarithmic differentiation.

Derivative of ex

If is a differentiable function of , then

u u

u x

d due e

dx dx

Example Derivative of ex

3

Find if xdyy e

dx

3

3

3

2

where

3

x

u

x

y e u x

dy due

dx dx

e x

Derivative of ax

If is a differentiable function of and for 0 and 1,

lnu u

u x a a

d dua a a

dx dx

Derivative of ln x If is a differentiable function of and 0,

1ln

u x u

d duu

dx u dx

Example Derivative of ln x 2Find if ln 3

dyy x

dx

2 2

22

2

2

ln 3 where 3

13

31

636 2

3

y x u x

dy x

dxx

xxx

xx

Derivative of logax

If is a differentiable function of and 0,

1log

lna

u x u

d duu

dx u a dx

Rule 10 Power Rule For Arbitrary Real Powers

1

If is a positive differentiable function of and is a

differentiable function of , and

.n n

u x n

x

d duu nu

dx dx

Example Power Rule For Arbitrary Real Powers

3 4If 5 , find

.

dyy x

dx

3

3

4 13

4 13

Using the Power Rule

4 5 5

5 4 5

dyx

dx

x

Logarithmic DifferentiationSometimes the properties of logarithms can be used to simplify the differentiation process, even if logarithms themselves must be introduced as a step in the process.

The process of introducing logarithms before differentiating is called logarithmic differentiation.

Example Logarithmic Differentiation

cosFind for xdyy x

dx

cos

cosLogs of both sides

Property of logs

Differentiate Implicitly

Product Rule

cos

ln ln

ln cos ln

ln cos ln

1 1sin ln cos

cossin ln

cossin ln

x

x

x

y x

y x

y x x

d dy x x

dx

x

dxdy

x x xy dx x

dy xy x x

dx x

dy xx x

sx x

Substitute for y

In ReviewRule Example

dx

duee

dx

d uu

dx

duaaa

dx

d uu ln

dx

du

uu

dx

d

1ln

dx

du

auu

dx

da

ln

1log

xxx eeedx

d 444 44

2623

123ln

22

x

xxxx

dx

d

322ln22 33 22

xdx

d xxxx

3210ln3

13log

22

x

xxxx

dx

d

Section 3.9 – Derivatives of Exponential and Logarithmic Functions Derivation of this derivative:

x

x

x

eydx

dy

dx

dy

y

xexey

ey

11

lnlnln

Let’s use these formulas!

Using find

Using find

xxedx

d 62

dx

duee

dx

d uu

dx

duaaa

dx

d uu ln)( 25 xdx

d

At what point on the graph of the function y=3t – 4 does the tangent line have slope 12?

How can we find this out? Find dy/dt and set it equal to zero. Solve for t. Evaluate f(t) in the original equation. Hint: t is a messy number, store value for accuracy in final answer.

A line with slope m=1/a passes through the origin and is tangent to the graph of y = ln x. What is the value of m?

We know Point of tangency has coordinates (a, ln a) for some positive a Tangent line has slope m = 1/a Tangent line passes through the origin so equation has point (0, 0)

Find slope with formula and set it equal to 1/a. Solve for a.

Sometimes we should use logarithms to simplify before differentiation.

Find dy/dx for y = x x

Use logs to lose the exponent, THEN take the derivative implicitly.

How Fast does H1N1 Spread?The spread of a flu in a certain school is modeled by the equation

Where P(t) is the total number of students infected t days after the fluwas first noticed. Many of them may already be well again at time t.

a) Estimate the initial number of students infected with the flu.b) How fast is the flu spreading after 3 days?c) When will the flu spread at its maximum rate? What is this rate?

tetP

31

100)(