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The Quotient Rule: Derivative Fraction Guide

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'

Formula

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}

Easy Example Problem

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}
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