Derivatives of Exponential Functions

On this page we'll consider how to differentiate exponential functions.

Exponential functions have the form f (x) = a^x, where a is the base. The base is always a positive number not equal to 1.

If the base is equal to the number e:

then the derivative is given by

(This formula is proved on the page Definition of the Derivative.)

The function $$ is often referred to as simply the exponential function.

Besides the trivial case $$ the exponential function $$ is the only function whose derivative is equal to itself.

Now we consider the exponential function $$ with arbitrary base $$ $$ and find an expression for its derivative.

As $$ then

Using the chain rule, we have

Thus

In the examples below, determine the derivative of the given function.

Solved Problems

Solution.

By the chain rule, we obtain:

Solution.

Solution.

Using the chain rule, we get

Solution.

By the chain rule,

Solution.

Using the chain rule, we have

Solution.

By the chain rule,