Sum Rule Calculator

Published on: April 20, 2025

Final Answer: Free • Full Steps: Plus

This Sum Rule Calculator helps you differentiate sums such as x^2 + x^3, x + 5, or 2x^3 + 5x^2. The sum rule says that the derivative of a sum is the sum of the derivatives.

Step-by-step method

Set up the terms in the sum.
Write the sum rule formula.
Split the derivative across the addition signs.
Differentiate each separated term as Step 4a, Step 4b, Step 4c, and so on.
Combine the term derivatives and simplify.

Formula: This is the sum rule formula.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

The sum rule lets you split one derivative across addition signs. After splitting the derivative, each term can be differentiated using rules such as the constant rule, variable rule, constant multiple rule, and power rule.

Example 1:

\(f\left(x\right) = x^{2} + x^{3}\)

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2}\), \(x^{3}\).

\(f\left(x\right) = x^{2} + x^{3}\)

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dx}\left(x^{2} + x^{3}\right) = \frac{d}{dx}\left(x^{2}\right) + \frac{d}{dx}\left(x^{3}\right)\)

Step 4a - Differentiate the separated term \(x^{2}\).

In this problem: The separated term is \(x^{2}\). Use the power rule to get its derivative.

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

Problem

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

Approach

Repeatable 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

\(2 x\)

Step 4b - Differentiate the separated term \(x^{3}\).

In this problem: The separated term is \(x^{3}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(x^{3}\right) = 3 x^{2}\)

Problem

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

Approach

Repeatable 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 \(x^{3}\). This is a power of \(x\), so the exponent is \(n = 3\).

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

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, \(n = 3\). Substitute this exponent into the power rule, but do not simplify yet.

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

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 \(3{x}^{2}\).

\(3{x}^{3 - 1} = 3{x}^{2}\)

Final answer

\(3 x^{2}\)

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(x\right) = 2 x + 3 x^{2} = x \left(3 x + 2\right)\)

Final answer:

\(f'\left(x\right) = x \left(3 x + 2\right)\)

Example 2:

\(f\left(x\right) = 2 x^{3} + 5 x^{2}\)

Step 1 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(2 x^{3}\), \(5 x^{2}\).

\(f\left(x\right) = 2 x^{3} + 5 x^{2}\)

Step 2 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

Step 3 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dx}\left(2 x^{3} + 5 x^{2}\right) = \frac{d}{dx}\left(2 x^{3}\right) + \frac{d}{dx}\left(5 x^{2}\right)\)

Step 4a - Differentiate the separated term \(2 x^{3}\).

In this problem: The separated term is \(2 x^{3}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(2 x^{3}\right) = 6 x^{2}\)

Problem

\(f\left(x\right) = 2{x}^{3}\)

Approach

Repeatable 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 \(2 x^{3}\). This is a constant multiple of a power of \(x\). The coefficient is \(2\), the power part is \(x^{3}\), and the exponent is \(n = 3\).

\(f\left(x\right) = 2{x}^{3}\)

Step 2

Step 2 - Apply the constant multiple rule first.

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

\(\frac{d}{dx}2{x}^{3} = 2\cdot \frac{d}{dx}{x}^{3}\)

Problem

\(f\left(x\right) = 2 x^{3}\)

Approach

Repeatable method

Set up the coefficient and the variable part.
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 \(2 x^{3}\). The constant coefficient is \(2\), and the variable part is \(x^{3}\).

\(2 x^{3} = 2\cdot \left(x^{3}\right)\)

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 \(2\) outside the derivative and leave \(x^{3}\) inside the derivative.

\(\frac{d}{dx}\left(2 x^{3}\right) = 2\cdot \frac{d}{dx}\left(x^{3}\right)\)

Final answer

\(2\cdot \frac{d}{dx}\left(x^{3}\right)\)

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 \(2\) outside.

\(2\cdot \frac{d}{dx}{x}^{3} = 2\cdot 3{x}^{3 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x^{3}\), multiply by the constant \(2\), and keep the solving line in power form. This gives \(6{x}^{2}\).

\(2\cdot 3{x}^{2} = 6{x}^{2}\)

Final answer

\(6 x^{2}\)

Step 4b - Differentiate the separated term \(5 x^{2}\).

In this problem: The separated term is \(5 x^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(5 x^{2}\right) = 10 x\)

Problem

\(f\left(x\right) = 5{x}^{2}\)

Approach

Repeatable 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 \(5 x^{2}\). This is a constant multiple of a power of \(x\). The coefficient is \(5\), the power part is \(x^{2}\), and the exponent is \(n = 2\).

\(f\left(x\right) = 5{x}^{2}\)

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.

\(\frac{d}{dx}5{x}^{2} = 5\cdot \frac{d}{dx}{x}^{2}\)

Problem

\(f\left(x\right) = 5 x^{2}\)

Approach

Repeatable method

Set up the coefficient and the variable part.
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 \(5 x^{2}\). The constant coefficient is \(5\), and the variable part is \(x^{2}\).

\(5 x^{2} = 5\cdot \left(x^{2}\right)\)

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 \(x^{2}\) inside the derivative.

\(\frac{d}{dx}\left(5 x^{2}\right) = 5\cdot \frac{d}{dx}\left(x^{2}\right)\)

Final answer

\(5\cdot \frac{d}{dx}\left(x^{2}\right)\)

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, \(n = 2\). Substitute this exponent into the power rule, then keep the coefficient \(5\) outside.

\(5\cdot \frac{d}{dx}{x}^{2} = 5\cdot 2{x}^{2 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x^{2}\), multiply by the constant \(5\), and keep the solving line in power form. This gives \(10{x}^{1}\).

\(5\cdot 2{x}^{1} = 10{x}^{1} = 10 x\)

Final answer

\(10 x\)

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(x\right) = 6 x^{2} + 10 x = 2 x \left(3 x + 5\right)\)

Final answer:

\(f'\left(x\right) = 2 x \left(3 x + 5\right)\)