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IBMathsDPApplications & Interpretation (AI)HLRevision Notes5. CalculusFurther DifferentiationProduct Rule

Product Rule (DP IB Applications & Interpretation (AI)): Revision Note

Lucy Kirkham

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Product Rule

What is the product rule?

  • The product rule states ifspace y is the product of two functionsspace u left parenthesis x right parenthesis andspace v left parenthesis x right parenthesis then

    space y equals u v

    bold space 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

size 16px space size 16px y to the power of size 16px apostrophe size 16px equals size 16px u size 16px v to the power of size 16px apostrophe size 16px plus size 16px v size 16px u size 16px apostrophe

  • Final answers should match the notation used throughout 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)

      • 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 formula

Simplify the answer if straightforward to do so or if the question requires a particular form

  • In trickier problems chain rule may have to be used when finding u apostrophe and v apostrophe

Examiner Tips and Tricks

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

    • 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 chain rule may be required inside product rule

    • i.e.  chain rule may be needed to differentiate u and 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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    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.