Product Rule Calculator

Published on: April 27, 2025
Final Answer: Free • Full Steps: Plus

This Product Rule Calculator helps you differentiate products of two functions such as (x^2 + 1)(x^3 - 2x) or x^2 sin(x). The product rule says the derivative of a product is the derivative of the first factor times the second, plus the first factor times the derivative of the second.

Step-by-step method

  1. Set up the two factors f(x) and g(x).
  2. Write the product rule formula.
  3. Differentiate the first factor.
  4. Differentiate the second factor.
  5. Substitute into the product rule formula.
  6. Simplify the result.

Formula:

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

Example 1:

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

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 - 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 - 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 - 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 - 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 - 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:

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

Example 2:

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

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 - 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 - 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 - 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 - 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 - 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:

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