The quotient rule

In a nutshell

A quotient is the result of dividing one quantity by another. The quotient rule is used to differentiate functions that are fractions consisting of one function on the numerator and one function on the denominator. This is called the quotient of two functions.

Example 1

Identify the two functions that make up the following quotient:

$f(x)=\dfrac{3x^4-2x+1}{\sin(x)}$ 

Given that this is a quotient of functions, you look for a function $u(x)$ in the numerator and a function $v(x)$ in the denominator. So:

$u(x)=\underline{3x^4-2x+1}$ and $v(x)=\underline{\sin(x)}$ .

Example 2

Identify the two functions that make up the following quotient:

$f(x)=x^{-3}\ln(x)$ 

Since negative powers result in a reciprocal, you have that:

$f(x)=\dfrac{\ln(x)}{x^3}$ 

Thus, you have that the numerator function $u(x)$ and denominator function $v(x)$ are:

$u(x)=\underline{\ln(x)}$ and $v(x)=\underline{x^3}$ .

Note: The function in the second example could alternatively be treated as a product of $x^{-3}$ and $\ln(x)$ . Hence the product rule can be used to differentiate $f(x)$ .

The quotient rule

Function $f(x)$	Derivative $f'(x)$
$\dfrac{u(x)}{v(x)}$	$\dfrac{u'(x)v(x)-u(x)v'(x)}{v(x)^2}$

PROCEDURE

1.	Given a function $f(x)$ , identify the numerator function $u(x)$ and the denominator function $v(x)$ .
2.	Differentiate these two each with respect to $x$ , so you obtain $u'(x)$ and $v'(x)$ .
3.	Use the formula $f'(x)=\dfrac{u'(x)v(x)-u(x)v'(x)}{v(x)^2}$ . Note: The denominator of this is the square of the function $v(x)$ and not the composition of $v$ with $v$ .

Example 3

Differentiate the following function with respect to $x$ :

$f\left(x\right)=\frac{x^3+2}{2x^2}$

Identify the numerator function and denominator function:

$u(x)=x^3+2$ and $v(x)=2x^2$ .

Differentiate each of these with respect to $x$ :

$u'(x)=3x^2$ and $v'(x)=4x$ .

Compute the derivative using the quotient rule:

$f^\prime\left(x\right)=\frac{3x^2\cdot2x^2-\left(x^3+2\right)\cdot4x}{\left(2x^2\right)^2}$

Simplify:

$\begin{aligned}f'(x)=& \space\dfrac{6x^4-4x^4-8x}{4x^4}\\=&\space\dfrac{2x^4-8x}{4x^4}\\ =&\space \underline{\dfrac{x^3-4}{2x^3}}\end{aligned}$