# At a Glance - 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.

#### Example 1

There's a skill needed for integrals (we'll explain integrals later) that we'll consider now: thinking backwards. Instead of taking a function and figuring out its derivative, think about looking at a derivative and figuring out what sort of function it came from. Try this out when looking over solutions to derivatives. If |

#### Exercise 1

Find the derivative of the function *f*(*x*) = *x*^{2}, using the limit definition of the derivative.

#### Exercise 2

Find the derivative of the function *f*(*x*) = *x*^{3}, using the limit definition of the derivative.

#### Exercise 3

Find the derivative of the function *f*(*x*) = *x*^{4}, using the limit definition of the derivative.

#### Exercise 4

Now it's time for pattern-finding. We know the following functions and their derivatives:

*f*(*x*) = *x*^{2 }*f ' *(*x*) = 2*x*

*f*(*x*) = *x*^{3 }*f ' *(*x*) = 3*x*^{2}

*f*(*x*) = *x*^{4 }*f ' *(*x*) = 4*x*^{3}

What's the pattern?

#### Exercise 5

Find the derivative of the power function *f*(*x*) = *x*^{10}.

#### Exercise 6

Find the derivative of the power function *f*(*x*) = *x*^{85}.

#### Exercise 7

Find the derivative of the power function *k*(*x*) = *x*^{3.5}.

#### Exercise 8

Find the derivative of the power function *k*(*x*) = *x ^{–}*

^{6}.

#### Exercise 9

Find the derivative of the power function

.

#### Exercise 10

Find the derivative of the power function *h*(*x*) = *x*^{(π + e)}.

#### Exercise 11

Find the derivative of the power function

.

#### Exercise 12

Find the derivative of the power function

.

#### Exercise 13

What is the derivative of the power function *k*(*x*) = *x*^{0}.

#### Exercise 14

Find the derivative of the power function

.

#### Exercise 15

For the derivative *f ' *(*x*) = 3*x*^{2}, find a possible original function.

#### Exercise 16

For the derivative *f ' *(*x*) = 8*x*^{7}, find a possible original function.

#### Exercise 17

For the derivative *f'*(*x*) = –3*x*^{–4}, find a possible original function.

#### Exercise 18

For the derivative *g'*(*x*) = –9*x*^{–10}, find a possible original function.

#### Exercise 19

For the derivative

,

find a possible original function, *h*(*x*).