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

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

Lucy Kirkham

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Quotient Rule

What is the quotient rule?

  • The quotient rule states if space y is the quotient fraction numerator u left parenthesis x right parenthesis over denominator v left parenthesis x right parenthesis end fraction then

    space y equals u over v

     

    • 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

  • Final answers should match the notation used throughout 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 x

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

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

How do I use the quotient 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 (like they do for 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 andspace v apostrophe

 STEP 2

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

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

Simplify the answer if straightforward or if the question requires a particular form

  • In trickier problems chain rule may have to be used when findingspace u apostrophe andspace v apostrophe,

Examiner Tips and Tricks

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

    • 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)

  • 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 it can also be seen as a quotient rule question in disguise 

    • ... 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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    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.