3.9: Derivatives of Exponential and Logarithmic Functions, p. 172 AP Calculus AB/BC 3.9: Derivatives of Exponential and Logarithmic Functions, p. 172

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

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.

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

Example 1

Example 2

( and are inverse functions.) (chain rule)

( is a constant.) Incorporating the chain rule:

Example 3

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

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:

p

Example 4 = 1

Example 5 = 1

Example 6

Example 7 First, take the ln of both sides.

Example 7 (cont.)