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