# Trigonometry/Derivative of Inverse Functions

The inverse functions sin^{-1}(x), etc. have derivatives that are purely algebraic functions.

If y = sin^{-1}(x) then x = sin(y) and

So

Similarly,

If y = tan^{-1}(x) then x = tan(y) and

So

If y = sec^{-1}(x) then x = sec(y) and

So

## Power series[edit]

The above results provide an easy way to find the power series expansions of these functions.

This is uniformly convergent if |x| < 1 so can be integrated term by term. The constant of integration is zero since sin^{-1}(0) = 0, so

This is uniformly convergent if |x| < 1 so can be integrated term by term. The constant of integration is zero since tan^{-1}(0) = 0, so

Note that sec^{-1}(x) has no power series expansion about x=0, as it is not defined for x < 1 and has an infinite derivative when x = 1. An expansion about any point x = a > 1 in powers of (x-a) can be found uding Taylor's theorem; it will converge for 1 < x < 2a-1.