Grade 12 · Mathematics
Chain rule
Why it matters
Composite functions.
Advanced20m readingPrerequisite: Power & product rules
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.
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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.