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At a Glance - Power Functions

A power function is any function of the form f(x) = xa, 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) = x1/2

or

f(x) = x.5.

The function

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

g(x) = x6.

Sample Problem

The function 

f(x) = xx

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) = x0 = 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) = xa, the power rule tells us that

f '(x) = axa – 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 x2 is 2x, x1.5 is 1.5x0.5, and xπ is πxπ – 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 f ' (x) = 5x4, what could the original function f(x) be?


Exercise 1

Find the derivative of the function f(x) = x2, using the limit definition of the derivative. 


Exercise 2

Find the derivative of the function f(x) = x3, using the limit definition of the derivative. 


Exercise 3

Find the derivative of the function f(x) = x4, 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) = x2   f ' (x) = 2x

f(x) = x3   f ' (x) = 3x2

f(x) = x4   f ' (x) = 4x3

What's the pattern?


Exercise 5

Find the derivative of the power function f(x) = x10.


Exercise 6

Find the derivative of the power function f(x) = x85.


Exercise 7

Find the derivative of the power function k(x) = x3.5.


Exercise 8

Find the derivative of the power function k(x) = x6.


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) = x0.


Exercise 14

Find the derivative of the power function

.


Exercise 15

For the derivative f ' (x) = 3x2, find a possible original function. 


Exercise 16

For the derivative f ' (x) = 8x7, find a possible original function. 


Exercise 17

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


Exercise 18

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


Exercise 19

For the derivative 

,

find a possible original function, h(x). 


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