We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.

Exponential Functions

A power function has a variable x in the base and a constant for the power. An exponential function has a constant for the base and a variable for the power:

f(x) = ax.

In order to make life easier (we do that sometimes) we assume a is not 0, 1, or negative. If a is 0 then our function is f(x) = 0x, which is undefined when x = 0 and is 0 everywhere else. Not too interesting. 

If a is 1 then our function is f(x) = 1x = 1, which is a constant function, so also not too interesting. If a is negative, the function f(x) = ax is too weird to deal with: f will be negative some places, positive some places, and undefined at a lot of places (such as when x = 0.5.).

Sample Problem

The following are exponential functions:

f(x) = 2x
f(x) = ex
f(x) = (0.5)x

Sample Problem

The function

f(x) = xx

is not an exponential function because it has a variable for the base and the power.

When we think of "exponential function," a good default function to think of is

f(x) = ex

If we look at the estimates from the previous exercise, we're estimating that when f(x) = ex,

It turns out this is accurate. The derivative of the function f(x) = ex is

f'(x) = ex.

f(x) = ex is its own derivative.

Sample Problem

If f(x) = ex then f ' (4) = e4.

Of course, there are exponential functions with other bases besides e. We'll give the derivative rule here, and the reasons after we talk about the chain rule.

If f(x) = ax, then f ' (x) = ax ln a.

Sample Problem

If f(x) = 2x, then f ' (x) = 2x ln 2. To find the value of the derivative at a specific value of x, we plug that value in for x in the derivative function:

f ' (4) = 24 ln 2 = 16ln2.

People who Shmooped this also Shmooped...