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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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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 bold italic y bold equals bold italic u over bold italic v where u and v are functions of x then the quotient rule is:
fraction numerator d y over denominator d x end fraction equals fraction numerator v fraction numerator d u over denominator d x end fraction minus u fraction numerator d v over denominator d x end fraction over denominator v squared end fraction

  • In function notation, if bold f bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold g bold left parenthesis bold italic x bold right parenthesis over denominator bold h bold left parenthesis bold italic x bold right parenthesis end fraction then the quotient rule can be written as:
straight f apostrophe left parenthesis x right parenthesis equals fraction numerator straight h left parenthesis x right parenthesis straight g apostrophe left parenthesis x right parenthesis minus straight g left parenthesis x right parenthesis straight h apostrophe left parenthesis x right parenthesis over denominator left parenthesis straight h left parenthesis x right parenthesis right parenthesis squared end fraction

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 bold f bold left parenthesis bold italic x bold right parenthesis bold equals bold g bold left parenthesis bold italic x bold right parenthesis bold left parenthesis bold h bold left parenthesis bold italic x bold right parenthesis bold right parenthesis to the power of bold minus bold 1 end exponent 
    • 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
      • bold g bold left parenthesis bold italic x bold right parenthesis bold left parenthesis bold h bold left parenthesis bold italic x bold right parenthesis bold right parenthesis to the power of bold minus bold 1 end exponent bold equals fraction numerator bold g bold left parenthesis bold italic x bold right parenthesis over denominator bold h bold left parenthesis bold italic x bold right parenthesis end fraction

Worked example

Quotient Rule Example, AS & A Level Maths revision notes

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    Previous:4.1.2 Product RuleNext:4.2.1 Implicit Differentiation
    Roger

    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.