Calculus – Power Rule, Sum Rule, Difference Rule



In this lesson, 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.

Definition of the Power Rule

The Power Rule of Derivatives gives the following:

For any real number n,

the derivative of f(x) = xn is f ’(x) = nxn-1

which can also be written as

Example:

Differentiate the following:
a) f(x) = x5
b) y = x100
c) y = t6

Solution:

a) f’’(x) = 5x4

b) y’ = 100x99

c) y’ = 6t5

Derivative of the function f(x) = x

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

The derivative of f(x) = x is f ’(x) = 1

which can also be written as

Example:

Differentiate f(x) = x

Solution:

f ’(x) = f ’(x1) = 1x0 = 1

Derivative of a Constant Function

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) = c is f ’(x) = 0

which can also be written as

Example:

Differentiate the following:
a) f(3)
b) f(157)

Solution:

a) f ‘(3) = f ‘(3x0) = 0(3x-1) = 0
b) f ‘(157) = 0

The Constant Multiple Rule

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 c is a constant and f is a differentiable function, then

Example:

Differentiate the following:
a) y = 2x4
b) y = –x

Solution:



The Sum Rule

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

If f and g are 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 5x2 + 4x + 7

Solution:



The Difference Rule

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

If f and g are both differentiable, then

 

Example:

Differentiate x8 – 5x2 + 6x

Solution:



Videos

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, I 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







Derivative Calculator

The following derivative calculator can show you the steps and rules used to get the derivative of the given function. Use it to check your answers.




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