1. Derivatives of Logarithmic Functions
Section 3.6

3. Example 1, differentiate

4. Example 2, differentiate
Find ln(sin x).
Solution:
Using the chain rule we have

5. Example 3, differentiate

6. Example 4, differentiate

7. Example 5, differentiate

8. Example 5, differentiate

9. The calculation of derivatives of complicated functions involving products, quotients, or powers can often be simplified by taking logarithms.
The method used in the next example is called logarithmic differentiation.

10. Example 6, differentiate
Solution:
Take log of both sides of the equation
Use the properties of logarithms to simplify:

11. Example 6, differentiate
Use implicit differentiation:

12. Example 6, differentiate
Substitute y back in:

14. 3.6 Derivatives of Logarithmic Functions
Summarize Notes
Read section 3.6
Homework
Pg.223 #2-32 (odd)

15. The Number e as a Limit
If f (x) = ln x, then f (x) = 1/x. Thus f (1) = 1. We now use this fact to express the number e as a limit
From the definition of a derivative as a limit, we have

16. Because f (1) = 1, we have
Then, by the continuity of the exponential function, we have