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Product Rule (Cambridge O Level Additional Maths)

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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 d y over denominator d x end fraction equals u fraction numerator d v over denominator d x end fraction plus v fraction numerator d u over denominator d 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:
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
     
  • 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:
y apostrophe space equals space u v apostrophe space plus space v u apostrophe
  

How do I know when to use the product rule?

  • The product rule is used when we are trying to differentiate the product of two functions
    • These can easily be confused with composite functions (see chain rule)
      • space sin left parenthesis cos space x right parenthesis is a composite function, “sin of cos of x”
      •  space sin space x cos space x is a product, “sin x times cos x”

How do I use the product rule?

  • Make it clear whatspace u comma space v comma space u apostrophe andspace v apostrophe are
    • arranging them in a square can help
      • opposite diagonals match up
 STEP 1
 Identify the two functions,space u andspace v
 Differentiate bothspace u andspace v with respect tospace x to findspace u apostrophe and v apostrophe

 STEP 2
Obtain fraction numerator straight d y over denominator straight d x end fraction by applying the product rule formulaspace fraction numerator straight d y over denominator straight d x end fraction equals u fraction numerator straight d v over denominator straight d x end fraction plus v fraction numerator straight d u over denominator straight d x end fraction
Simplify the answer if straightforward to do so or if the question requires a particular form

Product Rule Eg, AS & A Level Maths revision notes

 

Examiner Tip

  • The product rule formula is not on the list of formulas page – make sure you know it
  • 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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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.