Quotient Rule
What is the Quotient Rule?
The quotient rule differentiates a function that's the ratio of two other functions. It's commonly remembered with the mnemonic "low d-high minus high d-low, over low squared" — where "low" is the denominator g and "high" is the numerator f.
The formula requires g(x) ≠ 0 at the point of differentiation, since the original function itself would be undefined there.
What Each Variable Means
When to Use It
- Differentiating any expression written as one function divided by another, like x²/(x+1)
- Deriving tan x's derivative from sin x / cos x
- Anywhere a rate depends on the ratio of two changing quantities
Step-by-Step Examples
Example 1: Deriving tan x's derivative
Problem: Differentiate y = sin x / cos x
Split the quotient into numerator and denominator.
f = sin x, g = cos xDifferentiate each separately.
f' = cos x, g' = -sin xSubstitute into the quotient rule.
(cos x · cos x − sin x · (−sin x)) / cos²xThe Pythagorean identity simplifies the top to 1.
= 1 / cos²xExample 2: A rational function
Problem: Differentiate y = x² / (x + 1)
Split the quotient into numerator and denominator.
f = x², g = x + 1Differentiate each separately.
f' = 2x, g' = 1Substitute into the quotient rule.
((x+1)·2x − x²·1) / (x+1)²Distribute and collect like terms.
(2x² + 2x − x²) / (x+1)² = (x² + 2x) / (x+1)²Common Mistakes
Mistake: Reversing the order of subtraction in the numerator.
Fix: It's gf' − fg' (denominator times numerator's derivative, minus numerator times denominator's derivative) — not the other way around. Getting the order backwards flips the sign of the answer.
Mistake: Forgetting to square the denominator.
Fix: The denominator of the result is g², not just g — a very common dropped exponent.
Practice Questions
Differentiate y = (2x) / (x² + 1).
Hint: f = 2x, g = x² + 1; f' = 2, g' = 2x.
Differentiate y = (x − 1) / (x + 1).
Frequently Asked Questions
Can the quotient rule be derived from the product rule?
Yes — write f/g as f · g⁻¹, then apply the product rule together with the chain rule (since g⁻¹ is a composite function of g).
What happens if g(x) = 0?
The original function f/g is undefined at that point, so the quotient rule doesn't apply there either — check the domain before differentiating.
Related Formulas
Product Rule
Differentiates a product of two functions — you cannot simply multiply their individual derivatives.
Learn more →Chain Rule
Differentiates composite functions — functions nested inside other functions.
Learn more →Power Rule
The most fundamental differentiation rule — multiply by the exponent, then reduce the exponent by one.
Learn more →