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) = a^x.

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) = 0^x, which is undefined when x = 0 and is 0 everywhere else. Not too interesting. 

If a is 1 then our function is f(x) = 1^x = 1, which is a constant function, so also not too interesting. If a is negative, the function f(x) = a^x 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) = 2^x
f(x) = e^x
f(x) = (0.5)^x

Sample Problem

The function

f(x) = x^x

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) = e^x. 

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

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

f'(x) = e^x.

f(x) = e^x is its own derivative.

Sample Problem

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

Sample Problem

If f(x) = 2^x, then f ' (x) = 2^x 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.