Power Rule Notes

Published on: September 29, 2026

This Power Rule Calculator differentiates a term that is a variable raised to a power, and shows the working line by line. The old exponent drops down to become a coefficient and the exponent itself falls by one, which is the whole of the rule. A constant multiple in front simply rides along unchanged. Following the same pattern every time keeps the coefficient and the exponent from being mixed up.

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Formula:

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

See Example 1 Hide Example 1

Problem

\(f\left(x\right) = {x}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

\(f\left(x\right) = {x}^{2}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dx}{x}^{2} = 2{x}^{2 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(2{x}^{1}\).

\(2{x}^{2 - 1} = 2{x}^{1} = 2 x\)

Final Answer

\(f'\left(x\right) = 2 x\)

See Example 2 Hide Example 2

Problem

\(f\left(x\right) = 2{x}^{4}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(2 x^{4}\). This is a constant multiple of a power of \(x\). The coefficient is \(2\), the power part is \(x^{4}\), and the exponent is \(n = 4\).

\(f\left(x\right) = 2{x}^{4}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(2\) is a constant coefficient.

\(\frac{d}{dx}2{x}^{4} = 2\cdot \frac{d}{dx}{x}^{4}\)

Problem

\(f\left(x\right) = 2 x^{4}\)

Approach

Step-by-step method

Set up the problem.
Write the constant multiple rule formula.
Apply the constant multiple rule.

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(2 x^{4}\). The constant coefficient is \(2\), and the variable part is \(x^{4}\).

\(2 x^{4} = 2\cdot \left(x^{4}\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(2\) outside the derivative and leave \(x^{4}\) inside the derivative.

\(\frac{d}{dx}\left(2 x^{4}\right) = 2\cdot \frac{d}{dx}\left(x^{4}\right)\)

Final Answer

\(2\cdot \frac{d}{dx}\left(x^{4}\right)\)

Step 3

Step 3 - Write the power rule formula.

In this problem: Now use the power rule on the remaining power of \(x\).

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Here, \(n = 4\). Substitute this exponent into the power rule, then keep the coefficient \(2\) outside.

\(2\cdot \frac{d}{dx}{x}^{4} = 2\cdot 4{x}^{4 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x^{4}\), multiply by the constant \(2\), and keep the solving line in power form. This gives \(8{x}^{3}\).

\(2\cdot 4{x}^{3} = 8{x}^{3}\)

Final Answer

\(f'\left(x\right) = 8 x^{3}\)

See Example 3 Hide Example 3

Problem

\(f\left(x\right) = {x}^{6}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{6}\). This is a power of \(x\), so the exponent is \(n = 6\).

\(f\left(x\right) = {x}^{6}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 6\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dx}{x}^{6} = 6{x}^{6 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(6{x}^{5}\).

\(6{x}^{6 - 1} = 6{x}^{5}\)

Final Answer

\(f'\left(x\right) = 6 x^{5}\)

See Example 4 Hide Example 4

Problem

\(f\left(x\right) = 5{x}^{3}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(5 x^{3}\). This is a constant multiple of a power of \(x\). The coefficient is \(5\), the power part is \(x^{3}\), and the exponent is \(n = 3\).

\(f\left(x\right) = 5{x}^{3}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(5\) is a constant coefficient.

\(\frac{d}{dx}5{x}^{3} = 5\cdot \frac{d}{dx}{x}^{3}\)

Problem

\(f\left(x\right) = 5 x^{3}\)

Approach

Step-by-step method

Set up the problem.
Write the constant multiple rule formula.
Apply the constant multiple rule.

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(5 x^{3}\). The constant coefficient is \(5\), and the variable part is \(x^{3}\).

\(5 x^{3} = 5\cdot \left(x^{3}\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(5\) outside the derivative and leave \(x^{3}\) inside the derivative.

\(\frac{d}{dx}\left(5 x^{3}\right) = 5\cdot \frac{d}{dx}\left(x^{3}\right)\)

Final Answer

\(5\cdot \frac{d}{dx}\left(x^{3}\right)\)

Step 3

Step 3 - Write the power rule formula.

In this problem: Now use the power rule on the remaining power of \(x\).

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Here, \(n = 3\). Substitute this exponent into the power rule, then keep the coefficient \(5\) outside.

\(5\cdot \frac{d}{dx}{x}^{3} = 5\cdot 3{x}^{3 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x^{3}\), multiply by the constant \(5\), and keep the solving line in power form. This gives \(15{x}^{2}\).

\(5\cdot 3{x}^{2} = 15{x}^{2}\)

Final Answer

\(f'\left(x\right) = 15 x^{2}\)