Chain Rule Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Chain Rule.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The derivative of a composite function f(g(x)) equals f'(g(x)) \cdot g'(x): the derivative of the outer function evaluated at the inner, times the derivative of the inner.

Derivative of outside times derivative of inside. Unpack layers.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: To differentiate a composite function, multiply the derivative of the outer by the derivative of the inner.

Common stuck point: Identify 'inside' and 'outside' functions first, then apply.

Sense of Study hint: Write u = [inner function] on scratch paper, then differentiate the outer function with respect to u, and multiply by du/dx.

Worked Examples

Example 1

easy
Find the derivative of f(x) = (3x + 1)^4.

Solution

  1. 1
    Identify the outer function u^4 and the inner function u = 3x + 1.
  2. 2
    Apply the chain rule: \frac{d}{dx}[u^4] = 4u^3 \cdot \frac{du}{dx}.
  3. 3
    The derivative of the inner function: \frac{du}{dx} = 3.
  4. 4
    Combine: f'(x) = 4(3x+1)^3 \cdot 3 = 12(3x+1)^3.

Answer

f'(x) = 12(3x + 1)^3
The chain rule says: differentiate the outer function, keep the inner function, then multiply by the derivative of the inner function. Think of it as peeling layers.

Example 2

medium
Find the derivative of f(x) = (x^2 + 1)^5.

Example 3

hard
Find the derivative of f(x) = \sin(x^3).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the derivative of f(x) = (5x - 2)^3.

Example 2

medium
Find the derivative of f(x) = \sqrt{4x + 3}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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