The Quotient Rule
Simple Explanation
To differentiate a QUOTIENT (fraction) of two functions: (f/g)' = (f'·g − f·g') / g² — the derivative of the top times the bottom, minus the top times the derivative of the bottom, all divided by the bottom squared.
Why Do We Need It?
This handles any expression written as one function divided by another — a very common form, that the Power Rule alone cannot correctly differentiate.
Formula
The Quotient Rule
(f/g)' = (f'·g − f·g') / g²
The derivative of a quotient of two functions follows a specific pattern: (derivative of top × bottom, minus top × derivative of bottom), all divided by the bottom squared.
- f, g
- — two differentiable functions of x, with g(x) ≠ 0
When to use it: Whenever one function of x is divided by another.
Worked Example
Apply the Quotient Rule
Differentiate f(x) = x² / (x+1), using the Quotient Rule.
Why Does This Work?
The Quotient Rule can actually be derived from the Product Rule, by writing f/g as f·g⁻¹ and applying the Product Rule together with the Chain Rule (to differentiate g⁻¹) — after simplifying, the two approaches give exactly the same result.
Real-Life Example
Differentiating a ratio that changes over time
Population density is population ÷ land area, and BOTH the population and the developed land area of a city are changing over time.
The Quotient Rule correctly finds how the density itself is changing, accounting for both the numerator and denominator changing simultaneously.
Practice
Differentiate f(x) = x / (x+2) using the Quotient Rule, then evaluate f'(0).
HardCommon mistake
Getting the order of subtraction wrong in the numerator (computing uv'−u'v instead of u'v−uv') — the order matters, since subtraction is not commutative.
Quick Review
- (u/v)' = (u'v − uv') / v².
- The numerator's subtraction order matters: derivative-of-top first.
- Can be derived from the Product Rule applied to f·g⁻¹.