Calculus

Misapplying the Quotient Rule

Students take the derivative of the numerator and the denominator separately and put them in a new fraction.

Quick check

Why students make it

A fraction looks like it should behave component by component. Students confuse the quotient rule with a linear rule, but derivatives do not distribute over division.

The rule done right

For y=f(x)g(x)y=\frac{f(x)}{g(x)} with g(x)≠0g(x) \ne 0, the derivative is y′=g(x)f′(x)−f(x)g′(x)[g(x)]2y'=\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}. The denominator is squared, and the order in the numerator matters.

Related resources

derivativesquotient-rulerational-functions