Chain Rule Calculator

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

This Chain Rule Calculator differentiates a function sitting inside another function, and shows the working line by line. The outer function is differentiated with the inner one left untouched, and the result is then multiplied by the derivative of the inner function. Forgetting that second factor is the usual error. Following the same pattern every time keeps the outer and inner layers clearly separate.

Step-by-step method

  1. Set up the problem.
  2. Write the chain rule formula.
  3. Differentiate the outer function with respect to u.
  4. Differentiate the inner function with respect to the variable.
  5. Multiply the derivatives and substitute u back.

Formula:

\(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the chain rule formula.
  3. Differentiate the outer function with respect to u.
  4. Differentiate the inner function with respect to the variable.
  5. Multiply the derivatives and substitute u back.

Step 1

Step 1 - Identify the inner function u and the outer function.

In this problem: The inner function is \(u = 2 x + 3\) and the outer function is \(u^{5}\).

\(\begin{gathered} u = 2 x + 3 \\ y = u^{5} \end{gathered}\)

Step 2

Step 2 - Write the chain rule formula.

In this problem: Differentiate the outer function with respect to u, then multiply by the derivative of the inner function.

\(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)

Step 3

Step 3 - Differentiate the outer function with respect to u.

In this problem: Treat u as the variable: the derivative of \(u^{5}\) is \(5 u^{4}\).

\(\frac{dy}{du} = 5 u^{4}\)

Step 4

Step 4 - Differentiate the inner function with respect to the variable.

In this problem: The derivative of \(2 x + 3\) is \(2\).

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

Step 5

Step 5 - Multiply the derivatives and substitute u back.

In this problem: Multiply the two derivatives and replace u with \(2 x + 3\), then simplify.

\(y'\left(x\right) = 10 \left(2 x + 3\right)^{4}\)

Final Answer

\(f'\left(x\right) = 10 \left(2 x + 3\right)^{4}\)
See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the chain rule formula.
  3. Differentiate the outer function with respect to u.
  4. Differentiate the inner function with respect to the variable.
  5. Multiply the derivatives and substitute u back.

Step 1

Step 1 - Identify the inner function u and the outer function.

In this problem: The inner function is \(u = 3 x\) and the outer function is \(\sin{\left(u \right)}\).

\(\begin{gathered} u = 3 x \\ y = \sin{\left(u \right)} \end{gathered}\)

Step 2

Step 2 - Write the chain rule formula.

In this problem: Differentiate the outer function with respect to u, then multiply by the derivative of the inner function.

\(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)

Step 3

Step 3 - Differentiate the outer function with respect to u.

In this problem: Treat u as the variable: the derivative of \(\sin{\left(u \right)}\) is \(\cos{\left(u \right)}\).

\(\frac{dy}{du} = \cos{\left(u \right)}\)

Step 4

Step 4 - Differentiate the inner function with respect to the variable.

In this problem: The derivative of \(3 x\) is \(3\).

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

Step 5

Step 5 - Multiply the derivatives and substitute u back.

In this problem: Multiply the two derivatives and replace u with \(3 x\), then simplify.

\(y'\left(x\right) = 3 \cos{\left(3 x \right)}\)

Final Answer

\(f'\left(x\right) = 3 \cos{\left(3 x \right)}\)

Example Problems

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