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!
- their derivation involves use of the identity
- 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,
- Rewrite,
- 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
- This subtle but crucial difference can be seen in their graphs
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
- rewrite as the product ‘
- The expression
integrates to 
- The expression
integrates to 
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