1. Exponential and Logarithmic Functions
Mt. Rushmore, South Dakota
Derivatives of
Exponential and Logarithmic Functions

2. Look at the graph of
If we assume this to be true, then:
The slope at x=0 appears to be 1.
definition of derivative

3. Now we attempt to find a general formula for the derivative of using the definition.
This is the slope at x=0, which we have assumed to be 1.

4. What does this mean????

5. At each point P(c, ec) on the graph y = ex, the slope of the graph equals the value of the function ec.

6. is its own derivative!
If we incorporate the chain rule:
We can now use this formula to find the derivative of

7. ( and are inverse functions.)
(chain rule)

8. ( is a constant.)
Incorporating the chain rule:

9. So far today we have:
Now it is relatively easy to find the derivative of

11. To find the derivative of a common log function, you could just use the change of base rule for logs:
The formula for the derivative of a log of any base other than e is:

13. Example Solution Differentiate the function f(x) = x ln x.
f '(x) = x (1/x) + (ln x)(1)
= 1 + ln x

14. Example
Differentiate the function.

15. Solution

16. Example
Differentiate the function with respect to t.

17. Solution

18. The Chain Rule for Logarithmic Functions
If u(x) is a differentiable function of x, then remember:

19. Example
Differentiate the function.

20. Solution

21. The Chain Rule for Exponential Functions
If u(x) is a differentiable function of x, then remember:

22. Example
Differentiate the function.
Solution

23. Example
Differentiate the function.
Solution

24. Example
Differentiate the function.
Solution