CalcRef

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

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

Find d/dx[sin x / x].
  1. f = sin x, f' = cos x, g = x, g' = 1
  2. [cos x · x - sin x · 1] / x² = (x cos x - sin x)/x²

Answer: (x cos x - sin x)/x²

Related formulas

Product Rule

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