Constant Multiple Rule Calculator

Published on: April 20, 2025
Final Answer: Free • Full Steps: Plus

This Constant Multiple Rule Calculator helps you differentiate expressions where a constant is multiplied by a function of x, such as 2x^4 or 5x^3. The calculator shows the formula, method, working steps, and final answer.

Step-by-step method

  1. Set up the coefficient and the variable part.
  2. Write the constant multiple rule formula.
  3. Apply the constant multiple rule.

Formula: This is the constant multiple rule formula.

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

The constant multiple rule says a constant coefficient stays outside the derivative.

Example 1:

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

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 - 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 - 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:

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

Example 2:

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

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

In this problem: We are given \(- 3 x^{2}\). The constant coefficient is \(-3\), and the variable part is \(x^{2}\).

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

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 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(-3\) outside the derivative and leave \(x^{2}\) inside the derivative.

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

Final answer:

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