Calculus

Derivative: Chain Rule

Formula

d/dx [f(g(x))] = f'(g(x)) × g'(x)

What is Derivative: Chain Rule?

The chain rule is used to differentiate composite functions. You differentiate the outer function, keep the inner function intact, then multiply by the derivative of the inner function.

Variables

f
f — Outer function
g
g — Inner function

Worked Examples

Example 1

Find the derivative of (x² + 1)³.

Solution:

Outer: u³ → 3u². Inner: x² + 1 → 2x. Result: 3(x² + 1)² × 2x = 6x(x² + 1)²

Frequently Asked Questions

Related Formulas

Derivative: Power Rule

d/dx [xⁿ] = n × xⁿ⁻¹

Derivative: Product Rule

(fg)' = f'g + fg'

Derivative: Quotient Rule

(f/g)' = (f'g − fg') / g²

Integral: Power Rule

∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)

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