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

2. HW Review p.181

3. 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

4. Thm- For any constant b > 0,
Thm- In particular,

5. 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

6. Derivative of ln x cont’d
Thm- For x > 0,

7. Example:
Find the derivative of f(x) = x ln x and g(x) = x 10^x

8. Other Base Logarithms
We can calculate the derivative of other base logs by using the change-of-base formula using ln x

9. Ex
Find the derivative of

10. Logarithmic Differentiation
Logarithmic Differentiation can be used in place of several product/quotient rules
Ex:

11. 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)

12. Ex
Use log differentiation

13. Ex 2
Differentiate using log dif.

14. Closure Find the derivative using logarithmic differentiation
HW: p.187 #
Test soon (Thurs?)

15. 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