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A LevelMaths: PureAQARevision Notes7. Differentiation7.3 Further Differentiation7.3.6 Differentiating Reciprocal and Inverse Trig Functions

Differentiating Reciprocal and Inverse Trig Functions (AQA A Level Maths: Pure)

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Differentiating Reciprocal and Inverse Trig Functions

How do I differentiate the reciprocal trigonometric functions?

  • The formulae for the derivatives of the reciprocal trigonometric functions are:
fraction numerator d over denominator d x end fraction left parenthesis sec x right parenthesis equals sec x tan x

fraction numerator d over denominator d x end fraction left parenthesis cosec x right parenthesis equals negative cosec x cot x

fraction numerator d over denominator d x end fraction left parenthesis cot x right parenthesis equals negative cosec squared x

  • You can derive the derivatives for sec, cosec, and cot using the chain rule and the derivatives of the basic trigonometric functions 

Diff Rec Inv Trig Illustr 1, AS & A Level Maths revision notes

How do I differentiate the inverse trigonometric functions?

  • The formulae for the derivatives of the inverse trigonometric functions are:
fraction numerator d over denominator d x end fraction left parenthesis arcsin x right parenthesis equals fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction

fraction numerator d over denominator d x end fraction left parenthesis arc cos x right parenthesis equals negative fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction

fraction numerator d over denominator d x end fraction left parenthesis arc tan x right parenthesis equals fraction numerator 1 over denominator 1 plus x squared end fraction

  • You can derive the derivatives of arcsin, arccos, and arctan using the reciprocal form of the chain rule and the derivatives of the basic trigonometric functions Diff Rec Inv Trig Illustr 3, AS & A Level Maths revision notes

Examiner Tip

  • The formulae for the derivatives of sec x, cosec x, and cot x are in the  formulae booklet – you don't need to memorise them
    • However, you should know how to derive the derivatives of sec, cosec, and cot using the chain rule
  • The formulae for the derivatives of arcsin, arccos, and arctan are not in the formulae booklet
    • You should know how to derive those derivatives using the reciprocal form of the chain rule

Worked example

Diff Rec Inv Trig Example, AS & A Level Maths revision notes

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    Roger

    Author: Roger

    Expertise: Maths

    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.