Differentiation Rules

The Chain Rule

Find the derivative of a composite function using the chain rule.

Example

Differentiate the outer function and multiply by the inner derivative.

highlighted = computed this step

Step 1 — Set up

Start with a composite function.

f(x)=sin⁡(x2)f(x)= \sin(x^{2})

Step 2 — Identify inner function

Identify the inner function.

u=x2u= \hlmath{x^{2}}

Step 3 — Differentiate outer

Differentiate the outer sine function.

ddusin⁡u=cos⁡(x2)\frac{d}{du}\sin u= \hlmath{\cos(x^{2})}

Step 4 — Differentiate inner

Differentiate the inner function.

u′=2xu'= \hlmath{2x}

Step 5 — Multiply the factors

Multiply outer derivative by inner derivative.

f′(x)=cos⁡(x2)⋅2xf'(x)= \hlmath{\cos(x^{2})\cdot 2x}
chain-rule If f(x) = outer(inner(x)), then f'(x) = outer'(inner(x))·inner'(x). Differentiate the outer function evaluated at the inner, then multiply by the inner's derivative.