Derivatives: rules and functions

Quotient rule

The quotient rule, (fg)′=f′g−fg′g2\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}, differentiates a fraction of functions.

(fg)′=f′g−fg′g2\begin{gathered} \left(\frac{f}{g}\right)' \\[4pt] =\frac{f'g-fg'}{g^2} \end{gathered}

The numerator is (derivative of the top) × (bottom) − (top) × (derivative of the bottom), and the denominator is squared.

(fg)′=f′g−fg′g2\begin{gathered} \left(\frac{f}{g}\right)' \\[4pt] =\frac{f'g-fg'}{g^2} \end{gathered}

The most common mistake is reversing the subtraction, so make it a habit to start with the derivative of the top.

Examples

  1. 01Differentiatexx+1\displaystyle \frac{x}{x+1}AnswerHide
    Answer1(x+1)2\displaystyle \frac{1}{(x+1)^2}
  2. 02Differentiatex+1x+2\displaystyle \frac{x+1}{x+2}AnswerHide
    Answer1(x+2)2\displaystyle \frac{1}{(x+2)^2}
  3. 03Differentiatexx+2\displaystyle \frac{x}{x+2}AnswerHide
    Answer2(x+2)2\displaystyle \frac{2}{(x+2)^2}
  4. 04Differentiatex+4x+5\displaystyle \frac{x+4}{x+5}AnswerHide
    Answer1(x+5)2\displaystyle \frac{1}{(x+5)^2}
  5. 05Differentiatex+3x+4\displaystyle \frac{x+3}{x+4}AnswerHide
    Answer1(x+4)2\displaystyle \frac{1}{(x+4)^2}
  6. 06Differentiatex+4x+2\displaystyle \frac{x+4}{x+2}AnswerHide
    Answer−2(x+2)2\displaystyle \frac{-2}{(x+2)^2}

FAQ

Q1Which order is the subtraction in the numerator?Show answerHide
A
(derivative of the numerator)×(denominator) − (numerator)×(derivative of the denominator). Reversing it flips the sign.

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Quotient rule
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Derivative of tan x

Published 2026-09-03 · Updated 2026-09-05