Chain Rule Calculator

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

This Chain Rule Calculator helps you differentiate composite functions such as (2x + 3)^5, sin(3x), or e^(x^2). The chain rule differentiates the outer function with respect to the inner function u, then multiplies by the derivative of the inner function.

Step-by-step method

  1. Identify the inner function u and the outer function.
  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:

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

Example 1:

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

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 - 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 - 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 - 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 - 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:

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

Example 2:

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

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 - 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 - 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 - 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 - 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:

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