Derivative of $\arcsin(x)$ I was trying to find the derivative of 
$$\arcsin(x) = \sin^{-1}(x)$$
I thought that I could use the rule of inversion:
$$({f^{-1}})'(x) = \dfrac{1}{f(x)'}$$
Therefor the derivative of $\arcsin(x)$ should be:
$$\dfrac{1}{\cos(x)}$$

But for some reason, this seems to only work for small $x$. Where did I do a mistake?
Greetings,
Finn

In fact, we have
$ (\arcsin x)' = \dfrac{1}{\sqrt{1-x^2}}$
since,$(f^{-1}(x))' = \dfrac 1{f'(y)}$
