Chain Rule Calculator

This Chain Rule Calculator helps you differentiate composite functions using a repeatable u-substitution pattern. Instead of memorising the rule blindly, it shows how to find the inside expression, call it u, rewrite the function in terms of u, differentiate both parts, and multiply them together.

Pattern to use every time

Use the chain rule when one expression is being placed inside another function. The repeated pattern is:

The something is the inside expression. Call it u.

Step-by-step method

1. Box the inside expression and call it u.
2. Rewrite the original function in terms of u.
3. Differentiate y with respect to u.
4. Differentiate u with respect to x.
5. Multiply dy/du by du/dx, then replace u with the original inside expression.
6. Write the final derivative.

Formula:

Example 1:

Step 1 - Box the inside expression and call it u.

In this problem: The inside expression is the part being acted on by the outside function.

Step 2 - Rewrite the original function in terms of u.

In this problem: Replace the inside expression with u so the outside function becomes easier to differentiate.

Step 3 - Differentiate y with respect to u.

In this problem: Now treat u like the variable and differentiate only the outside function.

Step 4 - Differentiate u with respect to x.

In this problem: Now differentiate the inside expression.

Step 5 - Multiply dy/du by du/dx, then replace u with the original inside expression.

In this problem: This sets up the chain rule product before the final simplification.

Step 6 - Write the final derivative.

In this problem: Simplify the product to get the final derivative.

Final answer:

Example 2:

Step 1 - Box the inside expression and call it u.

In this problem: The inside expression is the part being acted on by the outside function.

Step 2 - Rewrite the original function in terms of u.

In this problem: Replace the inside expression with u so the outside function becomes easier to differentiate.

Step 3 - Differentiate y with respect to u.

In this problem: Now treat u like the variable and differentiate only the outside function.

Step 4 - Differentiate u with respect to x.

In this problem: Now differentiate the inside expression.

Step 5 - Multiply dy/du by du/dx, then replace u with the original inside expression.

In this problem: This sets up the chain rule product before the final simplification.

Step 6 - Write the final derivative.

In this problem: Simplify the product to get the final derivative.

Final answer: