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

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.

What does this mean????

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

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

( and are inverse functions.) (chain rule)

( is a constant.) Incorporating the chain rule:

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:

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

Example Differentiate the function.

Solution

Example Differentiate the function with respect to t.

Solution

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

Example Differentiate the function.

Solution

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

Example Differentiate the function. Solution

Example Differentiate the function. Solution

Example Differentiate the function. Solution