Quotient Rule

Unit 2 · Differentiation — Rules

The Quotient Rule

The Quotient Rule gives the derivative of a ratio of two functions. The mnemonic is: "lo d-hi minus hi d-lo, over lo squared" (lo · d(hi) − hi · d(lo)) / lo².

Quotient Rule

\frac{d}{dx}\!\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}

AP Tip: Alternative: rewrite f(x)/g(x) as f(x)·[g(x)]⁻¹ and use the Product Rule + Chain Rule when it simplifies things.

Type 1

Basic Quotient Rule

Identify f (numerator) and g (denominator), compute f′ and g′, then apply the formula.

Example 1

Find the derivative.

\frac{d}{dx}\!\left[\frac{x^2 + 3}{x - 1}\right]

Example 2

Find the derivative.

\frac{d}{dx}\!\left[\frac{\sin x}{x}\right]

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Type 2

Finding Horizontal Tangents

Set the derivative equal to zero. Since the denominator is always positive (or never zero), only the numerator needs to equal zero.

Example 3

Find all x where f has a horizontal tangent line.

f(x) = \frac{x^2 - 1}{x^2 + 1}

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Type 3

Quotient Rule + Chain Rule

When the numerator or denominator requires the chain rule, apply it inside the quotient rule.

Example 4

Find the derivative.

\frac{d}{dx}\!\left[\frac{e^x}{x^2 + 1}\right]

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← Product Rule Chain Rule →