Quotient Rule (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Quotient rule

What is the quotient rule?

• The quotient rule is a formula that allows you to differentiate a quotient of two functions

  • i.e. one function divided by another

• If $\mathit{y}\mathbf{=}\frac{\mathit{u}}{\mathit{v}}$ where u and v are functions of x then the quotient rule is:

$\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$

• In function notation, if $\mathbf{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}}{\mathbf{h}\mathbf{\left(}\mathit{x}\mathbf{\right)}}$ then the quotient rule can be written as:

$\mathrm{f}\text{'}\left(x\right)=\frac{\mathrm{h}\left(x\right)\mathrm{g}\text{'}\left(x\right)-\mathrm{g}\left(x\right)\mathrm{h}\text{'}\left(x\right)}{(\mathrm{h}{\left(x\right))}^2}$

• As with the product rule, ‘dash notation’ may be used to make remembering it easier

$y=\frac{u}{v}$

$y\text{'}=\frac{vu\text{'}-uv\text{'}}{v^2}$

• Final answers should match the notation used throughout the question

How do I know when to use the quotient rule?

• The quotient rule is used when trying to differentiate a fraction where both the numerator and denominator are functions of$x$

  • if the numerator is a constant, negative powers can be used

  • if the denominator is a constant, treat it as a factor of the expression\

How do I use the quotient rule?

• Make it clear what$u,v,u\text{'}$ and$v\text{'}$ are

  • arranging them in a square can help

    • opposite diagonals match up (like they do for product rule)

 STEP 1

 Identify the two functions,$u$ and$v$

 Differentiate both$u$ and$v$ with respect to$x$ to find$u\text{'}$ and$v\text{'}$

 STEP 2

Obtain $\frac{\mathrm{d}y}{\mathrm{d}x}$ by applying the quotient rule formula$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{v\frac{\mathrm{d}u}{\mathrm{d}x}-u\frac{\mathrm{d}v}{\mathrm{d}x}}{v^2}$ Be careful using the formula – because of the minus sign in the numerator, the order of the functions is important Simplify the answer if straightforward or if the question requires a particular form