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Introduction to Computing Derivatives - At A Glance:

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.

There's a skill needed for integrals 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.

Example 1

There's a skill needed for integrals 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 is 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

Find 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. 

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