The document discusses derivatives of exponential and logarithmic functions. It shows that the derivative of an exponential function e^x is itself, e^x. It also shows that the derivative of the natural logarithm function ln(x) is 1/x using the definition of derivative and the chain rule. Finally, it provides the general formula for deriving logarithmic functions of any base as -1/(xln(b)).

1. 3.9: Derivatives of Exponential and Logarithmic Functions Mt. Rushmore, South Dakota

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

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

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

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

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

10. 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:

11. 