OML Search

In these lessons, we will learn the Power Rule, the Constant Multiple Rule, the Sum Rule and the Difference Rule.

We have included a Derivative or Differentiation calculator at the end of the lesson. It can show the steps involved including the power rule, sum rule and difference rule.

Related Topics: More Calculus Lessons

It is not always necessary to compute derivatives directly from the definition. Instead, several rules have been developed for finding derivatives without having to use the definition directly. These formulas greatly simplify the task of differentiation.

The Power Rule of Derivatives gives the following:

For any real number

n,the derivative of

f(x) = xis^{n}f’(x) =nx^{n-1}which can also be written as

*Example:*

Differentiate the following:

a) *f*(*x*) = *x*^{5
}b) *y* = *x*^{100
}c) *y* = *t*^{6}

**Solution:**

a) *f*’’(*x*) = 5*x*^{4}

b) *y*’ = 100*x*^{99}

c) *y*’ = 6*t*^{5}

Using the power rule formula, we find that the derivative of the function ** f(x) = x** would be one.

The derivative of

f(x) =xisf’(x) = 1which can also be written as

*Example: *

Differentiate *f*(*x*) = *x*

*Solution: *

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

Using the power rule formula, we find that the derivative of a function that is a constant would be zero.

For any constant

c,The derivative of

f(x) =cisf’(x) = 0which can also be written as

** Example: **

Differentiate the following:

a) *f*(3)

b) *f*(157)

** Solution: **

a)* f* ‘(3) = *f* ‘(3*x*^{0}) = 0(3^{x-1}) = 0

b) *f* ‘(157) = 0

The constant multiple rule says that the derivative of a constant value times a function is the constant times the derivative of the function.

If

cis a constant andfis a differentiable function, then

*Example: *

Differentiate the following:

a) *y* = 2*x*^{4}

b) *y* = –*x*

*Solution:*

The Sum Rule tells us that the derivative of a sum of functions is the sum of the derivatives.

If

fandgare both differentiable, then

The Sum Rule can be extended to the sum of any number of functions.

For example (*f + g + h*)’ = *f’ + g’ + h’*

*Example: *

Differentiate 5*x*^{2} + 4*x* + 7

*Solution: *

The Difference Rule tells us that the derivative of a difference of functions is the difference of the derivatives.

If

fandgare both differentiable, then

*Example: *

Differentiate *x*^{8} – 5*x*^{2} + 6*x*

*Solution: *

Proof of the Power Rule for Derivatives

An explanation and some examples.

Differentiation Techniques: Constant Rule, Power Rule, Constant Multiple Rule, Sum and Difference Rule

Basic Derivative Rules - The Shortcut Using the Power Rule.

In this video, we look at finding the derivative of some very simple functions by using the power rule.

Power Rule and Derivatives, A Basic Example.

This video uses the power rule to find the derivative of a function.

Power Rule and Derivatives, Example #2.

This video uses the power rule to find the derivative of a function.

Power Rule and Derivatives, Example #3.

This video uses the power rule to find the derivative of a function.

Determining the derivatives of simple polynomials.

You can use the Mathway widget below to practice Calculus or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.