# 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) = e

^{4}.

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*ln

^{x }*a*.

### Sample Problem

If *f*(*x*) = 2* ^{x}*, then

*f '*(

*x*) = 2

*ln 2. To find the value of the derivative at a specific value of*

^{x}*x*, we plug that value in for

*x*in the derivative function:

*f ' *(4) = 2^{4 }ln 2 = 16ln2.