Derivative Proofs of Inverse Trigonometric Functions
To prove these derivatives, we need to know pythagorean identities for trig functions.
Proving arcsin(x) (or sin-1(x)) will be a good example for being able to prove the rest.
Derivative Proof of arcsin(x)
We know that
Taking the derivative of both sides, we get
We divide by cos(y)
Using a pythagorean identity for trig functions
We can substitute for cos(y)
Then we can substitute sin-1(x) back in for y and x for sin(y)
There you have it! The best part is, the other inverse trig proofs are proved similarly by using pythagorean identities and substitution, except the cofunctions will be negative.