3.9 Exponential and Logarithmic Derivatives

Mon Nov 9Do Now

Find the derivatives of:

1) 2)

HW Review p.181 #1-35

Exponential + Logarithmic Functions

• Logarithmic and exponential functions are among the most common functions encountered in applications.

• Population curves consist of logarithmic functions, particularly the natural logarithm.

• Growth/Decay, business applications use exponential functions

• Thm- For any constant b > 0,

• Thm- In particular,

Derivative of Natural Log• To determine the derivative of the

natural logarithm, let’s take a look at the graph of lnx and its slopes

Derivative of ln x cont’d

• Thm- For x > 0,

Example:

• Find the derivative of f(x) = x ln x and g(x) = x 10^x

Other Base Logarithms

• We can calculate the derivative of other base logs by using the change-of-base formula using ln x

•

Ex

• Find the derivative of

You try

• Find the derivatives• 1)

• 2)

Logarithmic Differentiation

• Logarithmic Differentiation can be used in place of several product/quotient rules

• Ex:

Logarithmic Differentiation

• 1) Take ln of both sides• 2) Use log rules to separate each factor• 3) Differentiate both sides (chain rule)• 4) Multiply by f(x) (original)

Ex

• Use log differentiation

Ex 2• Differentiate using log dif.

Closure

• Find the derivative using logarithmic differentiation

• HW: p.187 #1-49 odds, 79• 3.7-3.9 Quiz Mon

3.7-3.9 HW/Quiz ReviewTues Nov 10

• Do Now• Find the derivative of each• 1)

• 2)

HW Review: p.187 #1-49 79

3.7-3.9 Review

• Chain Rule– May contain all old rules (product, quotient, trig,

etc)• Derivatives of Inverses

– Explicit Derivatives (switch variables and differentiate)

– Inverse Trig (1 of them)• Logarithmic and Exponential Derivatives

– Most likely be included in chain rule– Logarithmic differentiation technique

Closure

• Journal Entry: How useful is logarithmic differentiation? When would you use it? When wouldn’t you?

• Quiz Thurs up to section 3.9