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The product rule and the quotient rule

The product rule differentiates a product by taking the derivative of one factor times the other, plus the other way round. The quotient rule takes the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, and divides by the denominator squared. The minus sign makes the order matter. Write down u and v and the derivatives before you assemble them, so you do not swap the terms.

(uv)' = u'v + uv'

the product rule

\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

the quotient rule

Symbols

u,\ v

the factors, or numerator and denominator

Example

$(xe^x)' = e^x + xe^x = (1 + x)e^x$ , which is $1$ when $x = 0$ .

$\left(\dfrac{x}{x + 1}\right)' = \dfrac{1}{(x + 1)^2}$ , which is $0.25$ when $x = 1$ .

In the quotient rule there is a minus between the terms. If you swap them, the sign is wrong.

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Part of Mathematics S2: Differentiation rules in practice.