3.9 Exponential and Logarithmic Derivatives Thurs Oct 8 Do Now Find the derivatives of: 1) 2)

HW Review p.181

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

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 #3 13 19 23 27 47 59 77 3.7-3.10 Test soon (Thurs?)

3.7-3.9 Review Chain Rule Derivatives of Inverses 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