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Product Rule (Cambridge (CIE) AS Maths) : Revision Note

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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 Tips and Tricks

  • 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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