Revision NotesExam QuestionsFlashcardsPast PapersMock Exams
IBMathsDPAnalysis & Approaches (AA)SLRevision Notes5. CalculusFurther DifferentiationChain Rule

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

Lucy Kirkham

Last updated

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

Did this video help you?

Chain Rule

What is the chain rule?

  •  The chain rule states ifbold space bold italic y is a function ofbold space bold italic u andbold space bold italic u is a function ofbold space bold italic x then

    space y equals f left parenthesis u left parenthesis x right parenthesis right parenthesis

    space fraction numerator bold d bold italic y over denominator bold d bold italic x end fraction bold equals fraction numerator bold d bold italic y over denominator bold d bold italic u end fraction bold cross times 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

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

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

How do I know when to use the chain rule?

  •  The chain rule is used when we are trying to differentiate composite functions

    • “function of a function”

    • these can be identified as the variable (usuallyspace x) does not ‘appear alone’

      • space sin space x – not a composite function, x ‘appears alone’

      • sin left parenthesis 3 x plus 2 right parenthesis is a composite function; x is tripled and has 2 added to it before the sine function is applied

How do I use the chain rule?

 STEP 1

 Identify the two functions

 Rewrite y as a function ofspace u; space y equals f left parenthesis u right parenthesis

 Write u as a function ofspace x; space u equals g left parenthesis x right parenthesis

 STEP 2

Differentiate y with respect to u to getspace fraction numerator straight d y over denominator straight d u end fraction

Differentiate u with respect to x to getspace fraction numerator straight d u over denominator straight d x end fraction

 STEP 3

Obtain fraction numerator straight d y over denominator straight d x end fraction by applying the formulaspace fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator straight d y over denominator straight d u end fraction cross times fraction numerator straight d u over denominator straight d x end fraction and substitutespace u back in forspace g left parenthesis x right parenthesis

 

  • In trickier problems chain rule may have to be applied more than once

Are there any standard results for using chain rule?

  • There are five general results that can be useful

    • Ifsize 16px space size 16px y size 16px equals begin mathsize 16px style stretchy left parenthesis f open parentheses x close parentheses stretchy right parenthesis end style to the power of size 16px n thenspace fraction numerator straight d y over denominator straight d x end fraction equals n straight f to the power of apostrophe stretchy left parenthesis x stretchy right parenthesis straight f stretchy left parenthesis x stretchy right parenthesis to the power of n minus 1 end exponent

    • Ifspace y equals straight e to the power of space f stretchy left parenthesis x stretchy right parenthesis end exponent thenspace fraction numerator straight d y over denominator straight d x end fraction equals f to the power of apostrophe left parenthesis x right parenthesis e to the power of space f stretchy left parenthesis x stretchy right parenthesis end exponent

    • Ifspace y equals ln stretchy left parenthesis f open parentheses x close parentheses stretchy right parenthesis thenspace fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator f to the power of apostrophe stretchy left parenthesis x stretchy right parenthesis over denominator f stretchy left parenthesis x stretchy right parenthesis end fraction

    • Ifsize 16px space size 16px y size 16px equals size 16px sin begin mathsize 16px style stretchy left parenthesis f open parentheses x close parentheses stretchy right parenthesis end style thenspace fraction numerator straight d y over denominator straight d x end fraction equals f to the power of apostrophe stretchy left parenthesis x stretchy right parenthesis cos stretchy left parenthesis f open parentheses x close parentheses stretchy right parenthesis

    • Ifspace y equals cos stretchy left parenthesis f open parentheses x close parentheses stretchy right parenthesis thenspace fraction numerator straight d y over denominator straight d x end fraction equals negative f to the power of apostrophe stretchy left parenthesis x stretchy right parenthesis sin stretchy left parenthesis f open parentheses x close parentheses stretchy right parenthesis

Examiner Tips and Tricks

  • You should aim to be able to spot and carry out the chain rule mentally (rather than use substitution)

    • every time you use it, say it to yourself in your head
      “differentiate the first function ignoring the second, then multiply by the derivative of the second function"

Worked Example

a) Find the derivative ofspace y equals left parenthesis x squared minus 5 x plus 7 right parenthesis to the power of 7.

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

b) Find the derivative ofspace y equals sin left parenthesis straight e to the power of 2 x end exponent right parenthesis.

5-2-2-ib-sl-aa-only-chain-we-soltn-b

You've read 0 of your 5 free revision notes this week

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourselfFlashcards

    Did this page help you?

    Previous:Differentiating Special FunctionsNext:Product Rule
    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.

    OSZAR »