- Topics At a Glance
**Derivatives of Basic Functions**- Constant Functions
- Lines
**Power Functions**- Exponential Functions
- Logarithmic Functions
- Trigonometric Functions
- Derivatives of More Complicated Functions
- Derivative of a Constant Multiple of a Function
- Multiplication by -1
- Fractions With a Constant Denominator
- More Derivatives of Logarithms
- Derivative of a Sum (or Difference) of Functions
- Derivative of a Product of Functions
- Derivative of a Quotient of Functions
- Derivatives of Those Other Trig Functions
- Solving Derivatives
- Using the Correct Rule(s)
- Giving the Correct Answers
- Derivatives of Even More Complicated Functions
- The Chain Rule
- Re-Constructing the Quotient Rule
- Derivative of a
^{x} - Derivative of
*ln*x - Derivatives of Inverse Trigonometric Functions
- The Chain Rule in Leibniz Notation
- Patterns
- Thinking Backwards
- Implicit Differentiation
- Computing Derivatives Using Implicit Differentiation
- Using Leibniz Notation
- Using Lagrange Notation
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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

The following are all power functions:

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 ^{–}*

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.

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 |

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 is 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 ^{–}*

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*) = *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*) = 8x^{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.