# Power Functions

A **power function** is any function of the form *f*(*x*) = *x ^{a}*, where

*a*is any real number.

### Sample Problem

The following are all power functions:

### Sample Problem

The following are all power functions, written deceptively.The function

is a power function since it can be written as

*f*(*x*) = *x*^{1/2}

or

*f*(*x*) = *x*^{.5}.

The function

is also a power function, since this can be written as

*g*(*x*) = *x ^{–}*

^{6}.

### Sample Problem

The function

*f*(*x*) = *x ^{x}*

is not a power function, because the exponent is a variable instead of a constant.

We usually assume the exponent *a* isn't 0, because if *a* is 0 we find a power function. The function

*f*(*x*) = *x*^{0} = 1

is a constant function, and we already know how to deal with those.

Now that we've got an idea of what a power function is we can talk about their derivatives. Luckily, there's a handy rule we can use to find the derivative of *any *power function that we want.

## Power Rule

Given a power function, *f*(*x*) = *x ^{a}*, the

**power rule**tells us that

*f* '(*x*) = *ax ^{a – }*

^{1}

To find the derivative, just take the power, put in front and then subtract 1 from the power.

Using this rule, we can quickly find the derivative of any power function. The derivative of *x*^{2} is 2*x*, *x*^{1.5} is 1.5*x*^{0.5}, and *x*^{π} is π*x*^{π – 1}.

No matter the power function, we can find its derivative.