Edexcel A Level Further Maths: Core Pure:复习笔记5.2.4 Calculus involving Inverse Trig

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
    •  which is equivalent to
    •  which is equivalent to

What are the derivatives of the inverse trigonometric functions?

  • Unlike other derivatives these look completely unrelated at first
    • their derivation involves use of the identity
    • hence the squares and square roots!
  • All three are given in the formula booklet
  • Note with the derivative of  that  is the same as

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

  • For
    • Rewrite,
    • Differentiate implicitly,
    • Rearrange,
    • Using the identity  rewrite,
    • Since,
  • Similarly, for
  • Notice how the derivative of is positive but is negative for 
    • This subtle but crucial difference can be seen in their graphs
      • has a positive gradient for all values of  in its domain
      • has a negative gradient for all values of  in its domain

Exam Tip

  • For  the terms on the denominator can be reversed (as they are being added rather than subtracted)
    • 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  is


b) Find the derivative of .


Integrating Inverse Trig Functions

How do I integrate inverse trig functions?

  • Use integration by parts in the same way you would integrate
  • These can be integrated using parts however
    • rewrite as the product ‘’ and choose  and
    • 1 is easy to integrate and the inverse trig functions have standard derivatives listed in the formula booklet
  • The expression  integrates to
  • The expression  integrates to