Quotient Rule (Cambridge International IGCSE Further Maths)

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 $y=\frac{u}{v}$ 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 $f(x)=\frac{g(x)}{h(x)}$ then the quotient rule can be written as:

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

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

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

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

• 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$u,v,u'$ and$v'$ 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$u'$ and$v'$

STEP 2

Obtain $\frac{dy}{dx}$ by applying the quotient rule formula$\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$ 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

Exam Hint

• The quotient rule formula is not on the list of formulas page – you have to memorise it

  • however if you do forget it in an exam, you could rewrite $y=\frac{u}{v}$ as $y=u\times v^{-1}$ and then use the product rule
  • (the quotient rule is really doing exactly this)

• Be careful using the formula – because of the minus sign in the numerator the order of the functions is important!

Worked Example