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?
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- 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
- This subtle but crucial difference can be seen in their graphs
Examiner 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 .