Quotient Rule — Mathematics Knowledge System

Overview

The quotient rule gives the derivative of a ratio of two functions. It can be derived from the product rule combined with the chain rule, and it's essential whenever a function is written as f(x)/g(x) rather than a product.

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Formal Definition

Definition

For differentiable functions f and g, with g(x) ≠ 0:

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

Quotient Rule

Worked Examples

Let f=x²+1, g=x; f'=2x, g'=1.
$f'=2x,\ g'=1$
Apply the quotient rule.
$h'(x) = \frac{2x \cdot x - (x^2+1)\cdot 1}{x^2} = \frac{2x^2 - x^2 - 1}{x^2}$
Simplify the numerator.
$h'(x) = \frac{x^2 - 1}{x^2}$

Answer: h'(x) = (x² - 1)/x²

Practice Problems

Difficulty 3/10

Differentiate f(x) = (3x - 1)/(2x + 5).

Difficulty 4/10

Differentiate f(x) = e^x / x.

Common Mistakes

Common Mistake

Reversing the order of terms in the numerator: writing fg' - f'g instead of f'g - fg'.

Order matters because subtraction isn't commutative. Always keep the derivative of the numerator (f') multiplying g first: (f'g - fg')/g².

Common Mistake

Forgetting to square the denominator.

The denominator of the result is always [g(x)]², not g(x) alone.

Summary

Quotient rule: (f/g)' = (f'g - fg')/g².
Derivable from the product rule by writing f/g = f·g⁻¹ and applying the chain rule to g⁻¹.
Watch the order of subtraction in the numerator and remember to square the denominator.

References

WebsiteWikipedia — Quotient rule