---
title: "The Chain Rule"
description: "Differentiate a composition by multiplying the rate of the outer function by the rate of the inner one. The rule is what makes substitution, related rates and implicit differentiation possible."
canonical: https://lightmysky.com/learn/mathematics/the-chain-rule-mt_kdhl4dmwJn
source: https://lightmysky.com/learn/mathematics/the-chain-rule-mt_kdhl4dmwJn.md
retrieved: 2026-09-12
---

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# The Chain Rule

Differentiate a composition by multiplying the rate of the outer function by the rate of the inner one. The rule is what makes substitution, related rates and implicit differentiation possible.

Subject: Mathematics · Area: Calculus & Analysis · Ages 18 to 19
Page: https://lightmysky.com/learn/mathematics/the-chain-rule-mt_kdhl4dmwJn

## Ready when they can

- Differentiate (3x² + 1)⁵ and name the inner and outer functions
- Apply the rule twice on a triple composition
- Explain the rule as two rates multiplying, using units

## Lesson: The chain rule: rates times rates

For f(g(x)), differentiate the outside first with the inside untouched, then multiply by the derivative of the inside. In symbols: f'(g(x)) times g'(x). With (3x squared + 1)^5, the outside gives 5(3x squared + 1)^4 and the inside gives 6x, so the answer is 30x(3x squared + 1)^4.

**Example.** Three layers need the rule twice. For e^(sin(x squared)), work inward: keep e^(sin(x squared)), multiply by cos(x squared), then by 2x. The trailing 2x factor makes the gradient 0 at x = 0. Likewise sin(ln(x squared)) gives cos(ln(x squared)) times 2/x.

Rates multiply, and units show it: dy/dx times dx/dt gives dy/dt. In related rates, differentiate with respect to time before touching the numbers. For a balloon with V = (4/3) pi r cubed, dV/dt = 4 pi r squared times dr/dt, so r = 3 with dr/dt = 0.5 gives 18 pi.

**Tip.** Two slips spoil answers. Never drop the inner derivative, and keep the inside intact: cos(x squared) gives -2x sin(x squared), never -sin(2x). Check by differentiating your answer: the chain rule should bring back the start.

**Recap.** Outside first with inside untouched, times the inside derivative, once per layer.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [Composite Functions](https://lightmysky.com/learn/mathematics/composite-functions-mt_jqGJHXgYNw)
- [The Power Rule for Differentiating Polynomials](https://lightmysky.com/learn/mathematics/the-power-rule-for-differentiating-polynomials-mt_l7f4j44bR3)
- [The Product and Quotient Rules](https://lightmysky.com/learn/mathematics/the-product-and-quotient-rules-mt_NJ7INJ0jfv)

## Opens up

- [Derivatives of Exponential, Logarithmic and Trigonometric Functions](https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT)
- [The Multivariable Chain Rule](https://lightmysky.com/learn/mathematics/the-multivariable-chain-rule-mt_CHSHdNxPmJ)
- [Integration by Substitution](https://lightmysky.com/learn/mathematics/integration-by-substitution-mt_jH-j_XsdC7)