A LevelMaths: Pure 3CIERevision Notes4. Differentiation4.1 Further Differentiation4.1.3 Quotient Rule

Quotient Rule (CIE A Level Maths: Pure 3)

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Roger

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30 April 2023

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Quotient Rule

What is the quotient rule?

The quotient rule is a formula that allows you to differentiate a quotient of two functions (ie one function divided by another)
If $y = \frac{u}{v}$ where u and v are functions of x then the quotient rule is:

\frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{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}}

Quotient Rule Eg, AS & A Level Maths revision notes

Examiner Tip

The quotient rule formula is in the formulae booklet – you don't have to memorise it.
Be careful using the formula – because of the minus sign in the numerator the order of the functions is important.
Look out for functions of the form $f (x) = g (x) {(h (x))}^{- 1}$
- You could differentiate that using a combination of the chain rule and the product rule (and it can be good practice for you to try it!)
- But it can also be seen as just a quotient rule question in disguise
  - $g (x) {(h (x))}^{- 1} = \frac{g (x)}{h (x)}$

Worked example

Quotient Rule Example, AS & A Level Maths revision notes

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Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.