High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize AP Calculus (AB) Textbook > Differentiation 1: Basics & Fundamental Properties

The Quotient Rule

The Product RulePrevious Section
Derivatives of Trig FunctionsNext Section
Popular Courses
Find My Course
0:00 / 0:00

The Quotient Rule

Our next differentiation shortcut lets us take derivatives of the quotient (division) of two functions.

Quotient Rule

The derivative of a quotient  f(x)g(x)\displaystyle\ \frac{f(x)}{g(x)}, where f(x)f(x) and g(x)g(x) are differentiable functions and g(x)≠0g(x)\ne 0, is:
[f(x)g(x)]′=f′(x)g(x)−f(x)g′(x)[g(x)]2\boxed{\quad \left[\frac{f\left(x\right)}{g\left(x\right)}\right]'=\frac{f'\left(x\right)g\left(x\right)-f\left(x\right)g'\left(x\right)}{\left[g\left(x\right)\right]^2}\quad }


Wize Tip
A nice way to remember the Quotient Rule is: "low d hi, minus hi d low, over low squared".

0:00 / 0:00

Example: The Quotient Rule

Find the following derivative
ddx[3xx−2]\displaystyle \frac{d}{dx}\left[{\frac{3x}{x-2}}\right]

ddx[3xx−2]\displaystyle \frac{d}{dx}\left[\frac{3x}{x-2}\right]
=(x−2)×ddx[3x]−(3x)×ddx[x−2](x−2)2\displaystyle =\frac{\left(x-2\right)\times\frac{d}{dx}\left[3x\right]-\left(3x\right)\times\frac{d}{dx}\left[x-2\right]}{\left(x-2\right)^2}
=(x−2)(3)−(3x)(1)(x−2)2\displaystyle =\frac{\left(x-2\right)\left(3\right)-\left(3x\right)\left(1\right)}{\left(x-2\right)^2}
=3x−6−3x(x−2)2\displaystyle =\frac{3x-6-3x}{\left(x-2\right)^2}
=−6(x−2)2\displaystyle =-\frac{6}{\left(x-2\right)^2}
Suppose g(x)\displaystyle g\left(x\right) is differentiable, f(x)=1+x g(x)x2f\left(x\right)=\displaystyle \frac{1+x\ g\left(x\right)}{x^2}, f′(1)=2f'\left(1\right)=2, g(1)=1g\left(1\right)=1, what is g′(1)g'\left(1\right)?