Calculus involving Inverse Trig (Edexcel A Level Further Maths: Core Pure)

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Differentiating Inverse Trig Functions

What are the inverse trigonometric functions?

  • arcsin, arccos and arctan are functions defined as the inverse functions of sine, cosine and tangent respectively
    •  arcsin open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses equals straight pi over 3 which is equivalent to sin space open parentheses pi over 3 close parentheses equals fraction numerator square root of 3 over denominator 2 end fraction
    •  arctan left parenthesis negative 1 right parenthesis equals fraction numerator 3 pi over denominator 4 end fraction which is equivalent to tan open parentheses fraction numerator 3 pi over denominator 4 end fraction close parentheses equals negative 1

What are the derivatives of the inverse trigonometric functions?

  • f left parenthesis x right parenthesis equals arcsin space x
    • f apostrophe left parenthesis x right parenthesis equals fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction
  • f left parenthesis x right parenthesis equals arccos space x
    • f apostrophe left parenthesis x right parenthesis equals negative fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction
  • f left parenthesis x right parenthesis equals arctan space x
    • f apostrophe left parenthesis x right parenthesis equals fraction numerator 1 over denominator 1 plus x squared end fraction
  • Unlike other derivatives these look completely unrelated at first
    • their derivation involves use of the identity cos squared space x plus sin squared space x identical to 1
    • hence the squares and square roots!
  • All three are given in the formula booklet
  • Note with the derivative of arctan space x that open parentheses 1 plus x squared close parentheses is the same as open parentheses x squared plus 1 close parentheses

How do I show or prove the derivatives of the inverse trigonometric functions?

  • For y equals arcsin space x
    • Rewrite, sin space y equals x
    • Differentiate implicitly, cos space y fraction numerator straight d y over denominator straight d x end fraction equals 1
    • Rearrange, fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator cos space y end fraction
    • Using the identity cos squared space y identical to 1 minus sin squared space y rewrite, fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator square root of 1 minus sin squared space y end root end fraction
    • Since, sin space y equals xfraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction
  • Similarly, for y equals arccos space x
    • cos space y equals x
    • negative sin space y fraction numerator straight d y over denominator straight d x end fraction equals 1
    • fraction numerator straight d y over denominator straight d x end fraction equals negative fraction numerator 1 over denominator sin space y end fraction
    • fraction numerator straight d y over denominator straight d x end fraction equals negative fraction numerator 1 over denominator square root of 1 minus cos squared space y end root end fraction
    • fraction numerator straight d y over denominator straight d x end fraction equals negative fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction
  • Notice how the derivative of y equals arcsin space x is positive but is negative for y equals arccos space x
    • This subtle but crucial difference can be seen in their graphs
      • y equals arcsin space x has a positive gradient for all values of x in its domain
      • y equals arccos space x has a negative gradient for all values of x in its domain

Examiner Tip

  • For space f left parenthesis x right parenthesis equals arctan space x the terms on the denominator can be reversed (as they are being added rather than subtracted)
    • space f apostrophe left parenthesis x right parenthesis equals fraction numerator 1 over denominator 1 plus x squared end fraction equals fraction numerator 1 over denominator x squared plus 1 end fraction
    • Don't be fooled by this, it sounds obvious but on awkward "show that" questions it can be off-putting!

Worked example

a)       Show that the derivative of arctan space x is fraction numerator 1 over denominator 1 plus x squared end fraction

5-8-3-ib-hl-aa-only-we2a-soltn

b)
Find the derivative of arctan left parenthesis 5 x cubed minus 2 x right parenthesis.

5-8-3-ib-hl-aa-only-we2b-soltn

Integrating Inverse Trig Functions

How do I integrate inverse trig functions?

  • Use integration by parts in the same way you would integrate ln x
  • These can be integrated using parts however
    • rewrite as the product ‘1 cross times f left parenthesis x right parenthesis’ and choose u equals f left parenthesis x right parenthesis and fraction numerator d v over denominator d x end fraction equals 1
    • 1 is easy to integrate and the inverse trig functions have standard derivatives listed in the formula booklet
  • The expression fraction numerator x over denominator 1 plus x squared end fraction integrates to 1 half ln open parentheses 1 plus x squared close parentheses
  • The expression plus-or-minus fraction numerator x over denominator square root of 1 minus x squared end root end fraction integrates to negative-or-plus square root of 1 minus x squared end root

table row cell integral arcsin open parentheses x close parentheses d x end cell equals cell x arcsin open parentheses x close parentheses plus square root of 1 minus x squared end root plus c end cell row cell integral arc cos open parentheses x close parentheses d x end cell equals cell x arc cos open parentheses x close parentheses minus square root of 1 minus x squared end root plus c end cell row cell integral arc tan open parentheses x close parentheses d x end cell equals cell x arc tan open parentheses x close parentheses minus 1 half ln open parentheses 1 plus x squared close parentheses plus c end cell end table

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.