Inverse Function Theorem

We shall next study the derivatives of the inverse trigonometric functions. Here is a general theorem which tells us when the derivative of the inverse function exists and gives a rule for computing its value.

We omit the proof that g'(y) exists. Intuitively, the curve y = f(x) has a non-horizontal tangent line, so the curve x = g(y) should have a nonvertical tangent line and thus g'(y) should exist.

The Inverse Function Rule from Chapter 2 says that (ii) is true if we assume (i). The proof of (ii) from (i) is an application of the Chain Rule:

g(f(x)) = x, g'(f(x))f'(x) = 1, g'(y)f'(x) = 1, g'(y) = 1/(f'(x)).

The Inverse Function Theorem shows that all the inverse trigonometric functions have derivatives. We now evaluate these derivatives.