Quotient rule

Statement

Let $f$ and $g$ be real functions on an open interval $I$, both differentiable at $a \in I$, with $g(a) \ne 0$. Then $f/g$ is differentiable at $a$, and $$\left( \frac{f}{g} \right)'(a) = \frac{f'(a) g(a) - f(a) g'(a)}{g(a)^2}.$$

Remarks

Write $f/g = f \cdot (1/g)$, and use the product rule and the reciprocal rule. For example, it gives $(\tan x)' = 1/\cos^2 x$. Mathlib states it as HasDerivAt.div.

Explanations

L1 human

The rule gives the slope of a quotient from the values and slopes of the numerator and the denominator. A common memory aid is "low d-high minus high d-low, over low squared".