Mt Rushmore South Dakota Derivatives of Exponential and

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

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

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 • Differentiate the function f(x) = x ln x. Solution 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