A LevelMathsAQAPureRevision NotesDifferentiationFurther DifferentiationChain Rule

Chain Rule (AQA A Level Maths) : Revision Note

Roger B

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Roger B

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AQA Maths: PureOCR Maths A: PureEdexcel Maths: Pure

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Chain Rule

What is the chain rule?

  • The chain rule is a formula that allows you to differentiate composite functions

  • If y is a function of u, and u is a function of x, then the chain rule tells us that:

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

  •   Note that because u is a function of x, y is also a function of x

  • In function notation, if bold italic y bold equals bold f bold left parenthesis bold g bold left parenthesis bold italic x bold right parenthesis bold right parenthesis then the chain rule can be written as:

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

  This allows us to differentiate more complicated expressions:

Chain Rule Illustr 1_ex, A Level & AS Level Pure Maths Revision Notes
Chain Rule Illustr 2_ex, A Level & AS Level Pure Maths Revision Notes

What are the special cases of the chain rule?

  • It allows us to differentiate a function raised to a power:

    • fraction numerator d over denominator d x end fraction left parenthesis left parenthesis straight f left parenthesis x right parenthesis to the power of n right parenthesis equals n left parenthesis straight f left parenthesis x right parenthesis right parenthesis to the power of n minus 1 end exponent straight f apostrophe left parenthesis x right parenthesis

Chain Rule Illustr 3_ex, AS & A Level Maths revision notes

  •  It allows us to differentiate functions that are not given in the form y= f(x):

    • fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator fraction numerator d x over denominator d y end fraction end fraction 

Chain Rule Illustr 4_ex, AS & A Level Maths revision notes

  • There is a useful case which is used to integrate certain functions:

    • fraction numerator d over denominator d x end fraction left parenthesis ln left parenthesis straight f left parenthesis x right parenthesis right parenthesis equals fraction numerator straight f apostrophe left parenthesis x right parenthesis over denominator straight f left parenthesis x right parenthesis end fraction

Examiner Tips and Tricks

  • The chain rule formulae are not in the exam formulae booklet – you have to know them.

  • When using the chain rule be sure to keep your functions straight (ie which function is y and which is u, or which is f and which is g).

Worked Example

Chain Rule Example, A Level & AS Level Pure Maths Revision Notes

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Roger B

Author: Roger B

Expertise: Maths Content Creator

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

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