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

Quotient 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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Quotient Rule

What is the quotient rule?

  • The quotient rule states that if space y is a quotient of functions of x

    • i.e. if space y equals u over v where u equals u open parentheses x close parentheses and v equals v open parentheses x close parentheses

    • then

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

      • This is given in the formula booklet

  • In function notation this could be written

space y equals fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction

  • As with product rule, ‘dash notation’ may be used

y equals u over v

y apostrophe equals fraction numerator v u apostrophe minus u v apostrophe over denominator v squared end fraction

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 quotient rule?

  • The quotient rule is used when trying to differentiate a fraction where both the numerator and denominator are functions ofspace bold italic x

    • If the numerator is a constant, negative powers can be used

      • E.g. 2 over open parentheses 3 x minus 7 close parentheses squared equals 2 open parentheses 3 x minus 7 close parentheses to the power of negative 2 end exponent

        • That can be differentiated using the chain rule

    • If the denominator is a constant, treat it as a factor of the expression

      • E.g. open parentheses 3 x minus 7 close parentheses squared over 2 equals 1 half open parentheses 3 x minus 7 close parentheses squared

        • That can be differentiated using the chain rule or by expanding the brackets

Examiner Tips and Tricks

The quotient rule will still work if the numerator or denominator is a constant. It is just usually quicker to use another method.

How do I use the quotient rule?

  • STEP 1

    Identify the two functions,space u andspace v

    Differentiate bothspace u andspace v with respect tospace x to findspace u apostrophe andspace v apostrophe
     

  • STEP 2

    Obtain fraction numerator straight d y over denominator straight d x end fraction by applying the quotient rule formula

    Simplify the answer if straightforward 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 quotient 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 (v and u apostrophe,  u and v apostrophe)

  • Be careful using the formula – because of the minus sign in the numerator, the order of the functions is important

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

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

Examiner Tips and Tricks

Look out for functions of the form space y equals f left parenthesis x right parenthesis left parenthesis g left parenthesis x right parenthesis right parenthesis to the power of negative 1 end exponent.

These can be differentiated using a combination of chain rule and product rule (it would be good practice to try!).

But they can also be seen as a quotient rule question in disguise 

  • f open parentheses x close parentheses open parentheses g open parentheses x close parentheses to the power of negative 1 end exponent close parentheses equals fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction

...and vice versa!

  • A quotient could be seen as a product by rewriting the denominator as left parenthesis g left parenthesis x right parenthesis right parenthesis to the power of negative 1 end exponent

Worked Example

Differentiate size 16px space size 16px f begin mathsize 16px style stretchy left parenthesis x stretchy right parenthesis end style size 16px equals fraction numerator size 16px cos size 16px space size 16px 2 size 16px x over denominator size 16px 3 size 16px x size 16px plus size 16px 2 end fraction blank with respect to space x.

5-2-2-ib-sl-aa-only-quotient-we-soltn

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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.