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)

Prove

We know that

Taking the derivative of both sides, we get

We divide by cos(y)

Using a pythagorean identity for trig functions

pythagorean identity

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.