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Derivatives · Basic Rules

Quotient Rule

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

The derivative of a quotient: (derivative of top times bottom minus top times derivative of bottom) over bottom squared.

Conditions. g(x) ≠ 0
  • 2 worked examples
  • 24 in Derivatives

Worked examples

Find d/dx[x/(x+1)].
  1. f = x, f' = 1, g = x+1, g' = 1
  2. [1·(x+1) - x·1] / (x+1)² = 1/(x+1)²

Answer: 1/(x+1)²

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Find d/dx[sin x / x].

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Related formulas

Product Rule

See all 24 Derivatives formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Quotient Rule?

The derivative of a quotient: (derivative of top times bottom minus top times derivative of bottom) over bottom squared. It is one of the derivatives formulas in the CalcRef reference.

When does the Quotient Rule apply?

The Quotient Rule holds under this condition: g(x) ≠ 0.

How do you use the Quotient Rule?

Worked example. Find d/dx[x/(x+1)]. f = x, f' = 1, g = x+1, g' = 1. [1·(x+1) - x·1] / (x+1)² = 1/(x+1)². Answer: 1/(x+1)².

Practice

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