I am trying to find the general formula / taylor expansion for $\arcsin(x)$ at $x=0$. I am not allowed to integrate, otherwise I would have used the binomial theorem and integrated afterwards. I have proven a statement before, which says that if I have a taylor series $T_n$ for a function $f(x)$, then $T_n(x^2)$ is the series for $g(x) = f(x^2)$. So basically, I can just substitute $x^2$ in the taylor series.
I suppose that this could come in handy, since I could then just expand $\arcsin(\sqrt x)$, which would make the derivatives easier.
All in all, I am not sure, however, how to find the formula. I looked it up and when I start differentiating $\arcsin(\sqrt x)$ a lot, I can not really see any pattern. What can I do here?