Product Rule Calculator

Published on: April 27, 2025

Final Answer: Free • Full Steps: Plus

This Product Rule Calculator differentiates two functions multiplied together, and shows the working line by line. The derivative of a product is not the product of the derivatives, so each factor takes a turn being differentiated while the other is left alone, and the two results are added. Identifying the factors first is what keeps the terms straight. Following the same pattern every time stops the two halves being run together.

Step-by-step method

Set up the problem.
Write the product rule formula.
Differentiate the first factor.
Differentiate the second factor.
Substitute into the product rule formula.
Simplify the result.

Formula:

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

See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

Set up the problem.
Write the product rule formula.
Differentiate the first factor.
Differentiate the second factor.
Substitute into the product rule formula.
Simplify the result.

Step 1

Step 1 - Set up the two factors f(x) and g(x).

In this problem: The first factor is \(x\) and the second factor is \(x^{4} - x^{2} - 2\).

\(\begin{gathered} f\left(x\right) = x \\ g\left(x\right) = x^{4} - x^{2} - 2 \end{gathered}\)

Step 2

Step 2 - Write the product rule formula.

In this problem: The derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second.

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

Step 3

Step 3 - Differentiate the first factor.

In this problem: Differentiate \(x\) to get \(1\).

\(f'\left(x\right) = \frac{d}{dx}\left(x\right) = 1\)

Step 4

Step 4 - Differentiate the second factor.

In this problem: Differentiate \(x^{4} - x^{2} - 2\) to get \(4 x^{3} - 2 x\).

\(g'\left(x\right) = \frac{d}{dx}\left(x^{4} - x^{2} - 2\right) = 4 x^{3} - 2 x\)

Step 5

Step 5 - Substitute into the product rule formula.

In this problem: Substitute the factors and their derivatives into the formula.

\(y'\left(x\right) = \left(1\right) \cdot \left(x^{4} - x^{2} - 2\right) + \left(x\right) \cdot \left(4 x^{3} - 2 x\right)\)

Step 6

Step 6 - Simplify the result.

In this problem: Expand the products and combine like terms.

\(y'\left(x\right) = x^{4} - x^{2} + x \left(4 x^{3} - 2 x\right) - 2 = 5 x^{4} - 3 x^{2} - 2\)

Final Answer

\(f'\left(x\right) = 5 x^{4} - 3 x^{2} - 2\)

See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

Set up the problem.
Write the product rule formula.
Differentiate the first factor.
Differentiate the second factor.
Substitute into the product rule formula.
Simplify the result.

Step 1

Step 1 - Set up the two factors f(x) and g(x).

In this problem: The first factor is \(x^{2}\) and the second factor is \(\sin{\left(x \right)}\).

\(\begin{gathered} f\left(x\right) = x^{2} \\ g\left(x\right) = \sin{\left(x \right)} \end{gathered}\)

Step 2

Step 2 - Write the product rule formula.

In this problem: The derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second.

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

Step 3

Step 3 - Differentiate the first factor.

In this problem: Differentiate \(x^{2}\) to get \(2 x\).

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

Step 4

Step 4 - Differentiate the second factor.

In this problem: Differentiate \(\sin{\left(x \right)}\) to get \(\cos{\left(x \right)}\).

\(g'\left(x\right) = \frac{d}{dx}\left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}\)

Step 5

Step 5 - Substitute into the product rule formula.

In this problem: Substitute the factors and their derivatives into the formula.

\(y'\left(x\right) = \left(2 x\right) \cdot \left(\sin{\left(x \right)}\right) + \left(x^{2}\right) \cdot \left(\cos{\left(x \right)}\right)\)

Step 6

Step 6 - Simplify the result.

In this problem: Expand the products and combine like terms.

\(y'\left(x\right) = x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}\)

Final Answer

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

Example Problems

Click one to load it into the box and solve it.