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

PROCEDURE

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:

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