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ASMaths: Pure 2CIERevision Notes4. Differentiation4.1 Further Differentiation4.1.2 Product Rule

Product Rule (CIE AS Maths: Pure 2)

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Roger

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Product Rule

What is the product rule?

  • The product rule is a formula that allows you to differentiate a product of two functions
  • If bold italic y bold equals bold italic u bold cross times bold italic v where u and v are functions of x then the product rule is:
fraction numerator size 22px d size 22px y over denominator size 22px d size 22px x end fraction size 22px equals size 22px u fraction numerator size 22px d size 22px v over denominator size 22px d size 22px x end fraction size 22px plus size 22px v fraction numerator size 22px d size 22px u over denominator size 22px d size 22px x end fraction

  • In function notation, if 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 cross times bold h bold left parenthesis bold italic x bold right parenthesis then the product rule can be written as:
begin mathsize 22px style straight f apostrophe left parenthesis x right parenthesis equals straight g left parenthesis x right parenthesis straight h apostrophe left parenthesis x right parenthesis plus straight h left parenthesis x right parenthesis straight g apostrophe left parenthesis x right parenthesis end style
     
  • The easiest way to remember the product rule is, for bold italic y bold equals bold italic u bold cross times bold italic v where u and v are functions of x:
begin mathsize 22px style y apostrophe space equals space u v apostrophe space plus space v u apostrophe end style
Product Rule Eg, AS & A Level Maths revision notes

 

Examiner Tip

  • The product rule formulae are in the formulae booklet – make sure you know how to find them.
  • Don't confuse the product of two functions with a composite function:
    • The product of two functions is two functions multiplied together
    • A composite function is a function of a function

 Product Rule Prod Comp Illustr, AS & A Level Maths revision notes 

  • To differentiate composite functions you need to use the chain rule

Worked example

Product Rule Example, A Level & AS Level Pure Maths Revision Notes

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    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.