AP Calculus AB and BC glossary

Quotient rule

The quotient rule differentiates a fraction: the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the denominator squared. The order matters because of the subtraction.

ddx[f(x)g(x)]=g(x)f′(x)−f(x)g′(x)(g(x))2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{\left(g(x)\right)^2}

Unlike the product rule, the quotient rule is not symmetric, so swapping the two terms in the numerator flips the sign of the whole answer. That is the single most common error with it.

Often avoidable

When the denominator is a single power, rewriting as a negative exponent and using the power rule is faster and less error-prone than the quotient rule.

Appears in: Unit 2: Defining the Derivative