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I need to find formula for $n$-th derivative of $\arcsin^k(\frac{x}{2})$.

I have found formula for

$$\left(\arcsin\frac{x}{2}\right)^{(n)}=\frac{(-i)^{n-1}(n-1)!}{\left(4-z^2\right)^{n/2}}P_{n-1}\left(\frac{i z}{\sqrt{4-z^2}}\right)$$

Where $P_n$ - Legendre polynomial.

I have found on the net formula for $n$-th derivative of composite function, however it didn't help. How can I find required formula?

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possible duplicate of Nth derivative of $\tan^m x$ – 5xum Apr 7 '14 at 13:40
@5xum That page is about $\tan^m x$, as the title claims. I don't see an obvious reduction to $\arcsin^m x$, do you? – Mario Carneiro Apr 7 '14 at 13:57
@5xum Maybe you can explain how to do something similar to $tan^m x$? – Somnium Apr 7 '14 at 20:49

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