3.9 Exponential and Logarithmic Derivatives Mon Nov 9 Do 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, Quiz Mon

HW/Quiz Review Tues Nov 10 Do Now Find the derivative of each 1) 2)

HW Review: p.187 #

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