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The quotient rule

The quotient rule differentiates a quotient of two functions. You take the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, and divide by the denominator squared. The minus sign makes the order matter. For $x/(x+1)$ the numerator of the answer is 1, so the derivative is one over the denominator squared. Check the arithmetic by inserting a value.

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

the quotient rule

Symbols

$u$	the numerator
$v$	the denominator

Example

$\left(\dfrac{x}{x + 1}\right)' = \dfrac{1}{(x + 1)^2}$ . At $x = 1$ it is $0.25$ , and at $x = 0$ it is $1$ .

The minus sign sits between the terms. If you swap them, the sign is wrong.

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Part of Mathematics R1: Differentiation rules in practice.