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The Quotient Rule: Derivative Fraction Guide

The quotient rule says the derivative of f(x) over g(x) equals the bottom times the derivative of the top minus the top times the derivative of the bottom, all over the bottom squared. A common memory aid is low d-high minus high d-low, over the square of what is below.

The Quotient Rule is used to find the derivative of a function that is divided by another function (a fraction).

Option 1: The Product Rule Comparison

It is very similar to the Product Rule:

• For the Product Rule (multiplication), you add the terms:

A' · B + A · B'

• For the Quotient Rule, instead of adding those terms, you subtract them:

A' · B - A · B'

• Finally, you divide the entire numerator by the square of the denominator:

(A' · B - A · B') / B²

Option 2: The Classic Rhythm (Rhyme)

"Low d-high minus high d-low, over the square of what's below."

* Low: the denominator B

* High: the numerator A

* d-high/d-low: A' and B'

What is the quotient rule for derivatives?

ddx[f(x)g(x)]=f′(x)g(x)−f(x)g′(x)[g(x)]2⏟Textbook Formula(AB)′=A′B−AB′B2⏟A / B Version\begin{gathered} \underbrace{\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) {\color{#ef4444}-} f(x)g'(x)}{[g(x)]^2}}_{\text{Textbook Formula}} \\\\ \underbrace{\left(\frac{A}{B}\right)' = \frac{{\color{#34d399}A'}B {\color{#ef4444}-} A{\color{#eab308}B'}}{B^2}}_{\text{A / B Version}} \end{gathered}

Quotient rule worked example: x over x squared plus one

Differentiate the rational function y = x / (x² + 1).

1. Identify f(x) = x and g(x) = x² + 1.

2. Find f'(x) = 1 and g'(x) = 2x.

3. Apply the Quotient Rule:

y′=(1)(x2+1)−(x)(2x)(x2+1)2y' = \frac{(1)(x^2 + 1) - (x)(2x)}{(x^2 + 1)^2}

4. Simplify the numerator:

y′=x2+1−2x2(x2+1)2=1−x2(x2+1)2y' = \frac{x^2 + 1 - 2x^2}{(x^2 + 1)^2} = \frac{1 - x^2}{(x^2 + 1)^2}

Another worked example

Differentiate y = (2x plus 1) over x. Step 1: the bottom is x with derivative 1; the top is 2x plus 1 with derivative 2. Step 2: form bottom times d-top minus top times d-bottom, which is x times 2 minus (2x plus 1) times 1. Step 3: that numerator is 2x minus 2x minus 1, so the derivative is negative 1 over x squared.

Aligned with the College Board AP Calculus CED — Unit 2 (Differentiation: Definition and Fundamental Properties), Topic 2.9 (The Quotient Rule). College Board ↗

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