# 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 0 everywhere else. Not too interesting.

If *a* is 1 then our function is *f*(*x*) = 1^{x} = 1, which is a constant function, 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*= 1/2.).

### 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*(

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