Derivatives of an Arcsine Product

Problem

Let $f(x) = \dfrac{\arcsin x}{\sqrt{1 - x^2}}$ .

1. Prove that for $n \in \mathbb{N}^*$ ,

\left(1 - x^2\right)f^{(n+1)}(x) - (2n+1)x\,f^{(n)}(x) - n^2 f^{(n-1)}(x) = 0.

Find $f^{(n)}(0)$ .

Answer

Solution

Difficulty	9/10
Topics	calculus, Recursion, Induction

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