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Product and chain rule

When a function is built up from several functions, we need special rules to differentiate it. The product rule applies to a product of two functions, and the chain rule to a function inside another function.

(fg)' = f'g + fg'

the product rule

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

the quotient rule

\big(f(g(x))\big)' = f'(g(x))\cdot g'(x)

the chain rule

Symbols

f,\ g

functions of x

Example

$f(x)=x^2e^x$ : $f'(x) = 2xe^x + x^2e^x$ (product rule), so $f'(0)=0$ .

In the quotient rule the minus sign and

g^2

in the denominator matter - don't swap the terms.

Practise differentiation for free →

Standard derivatives →

Part of Calculus: Differentiation.