# Quotient Rule

A LevelAQAEdexcelOCRAQA 2022Edexcel 2022OCR 2022

## Quotient Rule

You could use the Product Rule here, but that might get a little messy and a bit laborious. Here’s another rule which saves us a lot of time and effort.

A Level

## Quotient Rule Formula

For a function $\textcolor{limegreen}{y} = f(\textcolor{blue}{x}) = \dfrac{u(\textcolor{blue}{x})}{v(\textcolor{blue}{x})}$, we have the derivative (with respect to $\textcolor{blue}{x}$) given by

$\dfrac{d\textcolor{limegreen}{y}}{d\textcolor{blue}{x}} = \dfrac{df(\textcolor{blue}{x})}{d\textcolor{blue}{x}} = \dfrac{v(\textcolor{blue}{x})u'(\textcolor{blue}{x}) - u(\textcolor{blue}{x})v'(\textcolor{blue}{x})}{(v(\textcolor{blue}{x}))^2}$

A Level
A Level

## Example 1: Using the Quotient Rule

Say we have a function $\textcolor{limegreen}{y} = \dfrac{e^\textcolor{blue}{x}}{\sin \textcolor{blue}{x}}$. Find $\dfrac{d\textcolor{limegreen}{y}}{d\textcolor{blue}{x}}$.

[3 marks]

Let $u(\textcolor{blue}{x}) = e^\textcolor{blue}{x}$ and $v(\textcolor{blue}{x}) = \sin \textcolor{blue}{x}$. Then

\begin{aligned}\dfrac{d\textcolor{limegreen}{y}}{d\textcolor{blue}{x}} &= \dfrac{e^\textcolor{blue}{x}\sin \textcolor{blue}{x} - e^\textcolor{blue}{x}\cos \textcolor{blue}{x}}{\sin ^2 \textcolor{blue}{x}}\\[1.2em]&=\dfrac{e^\textcolor{blue}{x}(\sin \textcolor{blue}{x} - \cos \textcolor{blue}{x})}{\sin ^2 \textcolor{blue}{x}}\\[1.2em]&=e^\textcolor{blue}{x}(\sin \textcolor{blue}{x} - \cos \textcolor{blue}{x}) \cosec^2 \textcolor{blue}{x} \end{aligned}

A Level

## Example 2: Using the Quotient Rule (with the Product Rule)

Let $\textcolor{limegreen}{y} = \dfrac{(\textcolor{blue}{x}^2 - 1)\ln \textcolor{blue}{x}}{\cos \textcolor{blue}{x}}$ where $\textcolor{blue}{x}$ is measured in radians. Find $\dfrac{d\textcolor{limegreen}{y}}{d\textcolor{blue}{x}}$ and verify that there is a stationary point at the point $(1, 0)$.

[5 marks]

Let $u(\textcolor{blue}{x}) = (\textcolor{blue}{x}^2 - 1)\ln \textcolor{blue}{x}$ and $v(\textcolor{blue}{x}) = \cos \textcolor{blue}{x}$. Also, set $a = \textcolor{blue}{x}^2 - 1$ and $b = \ln \textcolor{blue}{x}$. Then

$\dfrac{da}{d\textcolor{blue}{x}} = 2\textcolor{blue}{x}$ and $\dfrac{db}{d\textcolor{blue}{x}} = \dfrac{1}{\textcolor{blue}{x}}$

so

$\dfrac{du(\textcolor{blue}{x})}{d\textcolor{blue}{x}} = 2\textcolor{blue}{x}\ln \textcolor{blue}{x} + \dfrac{\textcolor{blue}{x}^2 - 1}{\textcolor{blue}{x}}$ and $\dfrac{dv(\textcolor{blue}{x})}{d\textcolor{blue}{x}} = -\sin \textcolor{blue}{x}$

This gives

$\dfrac{d\textcolor{limegreen}{y}}{d\textcolor{blue}{x}} = \dfrac{\cos \textcolor{blue}{x}\left( 2\textcolor{blue}{x}\ln \textcolor{blue}{x} + \dfrac{\textcolor{blue}{x}^2 - 1}{\textcolor{blue}{x}}\right) + (\sin \textcolor{blue}{x} (\textcolor{blue}{x}^2 - 1)\ln \textcolor{blue}{x})}{\cos ^2 \textcolor{blue}{x}}$

When $\textcolor{blue}{x} = 1$,

$\dfrac{d\textcolor{limegreen}{y}}{d\textcolor{blue}{x}} = \dfrac{\cos 1((2 \times 0) + 0) + \sin 1 \times 0 \times 0}{\cos ^2 1}$

$= 0$

So, we can confirm that there is a stationary point at $\textcolor{blue}{x} = 1$.

A Level

## Example Questions

Let $u(x) = 1$ and $v(x) = x^3$.

Then

$\dfrac{dy}{dx} = \dfrac{0 - 3x^2}{x^6}$

$= \dfrac{-3}{x^4}$

Let $u(x) = 2\sin x$ and $v(x) = \cos x$. Then

$\dfrac{df(x)}{dx} = \dfrac{2\cos x \cos x + 2\sin x \sin x}{\cos ^2 x}$

$= 2\sec ^2 x$, by the identity $\sin ^2 x + \cos ^2 x \equiv 1$.

We also have

$d\dfrac{(2\tan x)}{dx} = 2\sec ^2 x$

Therefore, $\dfrac{df(x)}{dx} = \dfrac{d\tan x}{dx}$

This is an example of the Uniqueness Theorem. (See Product Rule, Q2).

Set $u = x^3$ and $v = 3^x$.

Then

$u' = 3x^2$ and $v' = 3^x\ln 3$

Using the quotient rule, we have

$\dfrac{dy}{dx} = \dfrac{3^{x + 1}x^2 - x^3 3^x\ln 3}{3^{2x}}$

Set $x = 0$ to give

$\dfrac{dy}{dx} = \dfrac{(3 \times 0) - (0 \times 1 \times \ln 3)}{1}$

$= 0$

So, we can confirm there is a stationary point at the origin.

A Level

A Level

## You May Also Like...

### A Level Maths Revision Cards

The best A level maths revision cards for AQA, Edexcel, OCR, MEI and WJEC. Maths Made Easy is here to help you prepare effectively for your A Level maths exams.

£14.99

### A Level Maths – Cards & Paper Bundle

A level maths revision cards and exam papers for Edexcel. Includes 2022 predicted papers based on the advance information released in February 2022! MME is here to help you study from home with our revision cards and practise papers.

From: £22.99

### Transition Maths Cards

The transition maths cards are a perfect way to cover the higher level topics from GCSE whilst being introduced to new A level maths topics to help you prepare for year 12. Your ideal guide to getting started with A level maths!

£8.99