Constant Multiple Rule Calculator

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

This Constant Multiple Rule Calculator differentiates a function that has been scaled by a number, and shows the working line by line. The constant can be pulled outside the derivative and put back afterwards, because scaling a function scales its rate of change by exactly the same factor. Only the variable part is actually differentiated. Following the same pattern every time keeps the constant from being differentiated by mistake.

Step-by-step method

  1. Set up the problem.
  2. Write the constant multiple rule formula.
  3. Apply the constant multiple rule.

Formula:

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the constant multiple rule formula.
  3. 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

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

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the constant multiple rule formula.
  3. Apply the constant multiple rule.

Step 1

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

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 \(-3\) 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

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

Example Problems

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