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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes5. CalculusTechniques & Applications of DifferentiationProduct Rule

Product Rule (DP IB Analysis & Approaches (AA)): Revision Note

Lucy Kirkham

Written by: Lucy Kirkham

Reviewed by: Dan Finlay

Updated on

DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SLDP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SL

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Product Rule

What is the product rule?

  • The product rule states that ifspace y is the product of two functions of x 

    • i.e. if y equals u v where space u equals u left parenthesis x right parenthesis andspace v equals v left parenthesis x right parenthesis

    • then

      fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals bold italic u fraction numerator bold d bold italic v over denominator bold d bold italic x end fraction bold plus bold italic v fraction numerator bold d bold italic u over denominator bold d bold italic x end fraction 

      • This is given in the formula booklet

  • In function notation this could be written as

y equals f left parenthesis x right parenthesis g left parenthesis x right parenthesis

space fraction numerator straight d y over denominator straight d x end fraction equals f left parenthesis x right parenthesis g to the power of apostrophe stretchy left parenthesis x stretchy right parenthesis plus g left parenthesis x right parenthesis f to the power of apostrophe stretchy left parenthesis x stretchy right parenthesis

  • ‘Dash notation’ may be used as a shorter way of writing the rule

space y equals u v

space y apostrophe equals u v apostrophe plus v u apostrophe

Examiner Tips and Tricks

In an exam, your final answers should match the notation used by the question.

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)

      • sin left parenthesis cos x right parenthesis space is a composite function, “sin of cos of x”

      •  sin x cos x space is a product, “sin x times cos x”

How do I use the product rule?

  • 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 formula space 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

Examiner Tips and Tricks

Use u comma space v comma space u apostrophe and v apostrophe for the elements of the product rule

  • Make it clear whatspace u comma space v comma space u apostrophe andspace v apostrophe are

  • Lay them out in a 'square' (imagine a 2x2 grid)

  • Those that are paired together are then on opposite diagonals (u and v apostrophe, v and u apostrophe)

For trickier functions, the chain rule may be required inside the product rule

  • I.e.  the chain rule may be needed to differentiate u and/or v

Worked Example

a)       Find the derivative ofspace y equals straight e to the power of x sin space x.

5-2-2-ib-sl-aa-only-product-we-soltn-a

b)       Find the derivative ofsize 16px space size 16px y size 16px equals size 16px 5 size 16px x to the power of size 16px 2 size 16px space size 16px cos size 16px space size 16px 3 size 16px x to the power of size 16px 2.

5-2-2-ib-sl-aa-only-product-we-soltn-a-b

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    Author
    Reviewer
    Lucy Kirkham

    Author: Lucy Kirkham

    Expertise: Head of STEM

    Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels.Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.