Calculus 12 · Mathematics · Lesson 3.3
Chain rule
- Level
- Grade 12
- Reading time
- 13 min reading
- Review
- Last reviewed 2026-09-03
- Standard
- Accuracy standard
- Errata
Abstract
Composite functions.
What is the chain rule in calculus?
Short answer
The chain rule differentiates a composite function — a function inside another function. It states that if y = f(g(x)), then dy/dx = f′(g(x)) · g′(x): differentiate the outer function evaluated at the inner one, then multiply by the derivative of the inner function. For example, the derivative of (3x² + 1)⁵ is 5(3x² + 1)⁴ · 6x.
Concept preview
What you will be able to do
- 01Identify a composite function as an outer function applied to an inner function.
- 02Apply the chain rule to differentiate composite functions, including the general power rule.
- 03Apply the chain rule together with the power, product and quotient rules.
- 04Apply the chain rule in Leibniz notation and to repeated (nested) compositions.
- 05Diagnose a flawed chain-rule application, especially a missing inner derivative.
Many functions are built by feeding one function into another. The expression is not a plain power of the variable; it is the fifth-power function applied to the quantity . The expression is the square-root function applied to . Functions assembled this way are called composite functions, and the previous lessons' rules do not handle them directly: the power rule differentiates a power of the variable, but here the base is itself a function, not just . To differentiate a composite function we need one more rule, the chain rule, which is the most important and most frequently used differentiation rule of all, precisely because composite functions are everywhere once you start looking.
School Plan
Chain rule
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Related topics
Frequently asked questions
- When do you use the chain rule?
- Whenever you differentiate a function nested inside another, such as (2x + 1)³ or sin(x²).
- What is a composite function?
- A function applied to the result of another function, written f(g(x)) — an inner function inside an outer one.