Calculus – Power Rule

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. In this lesson, we will look at the Power Rule.

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) = nx^n-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(3^x-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

Determining the derivatives of simple polynomials

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