$N$th derivative of $\sqrt{1-x^{2}}$

Question

I was wondering if there was any formula for the nth derivative of $\sqrt{1-x^{2}}$ and by an easy extension $-\sqrt{1-x^{2}}$. What would be even better is a Taylor series for the function at point $a$.

Thanks in advance.

Answer 1

At https://math.stackexchange.com/a/4657792, there is an answer: \begin{equation*} \bigl(\sqrt{1-x^2}\,\bigr)^{(k)} =-\sqrt{1-x^2}\,\frac{k!}{(2x)^k} \sum_{j=0}^{k}\frac{2^{j}(2j-3)!!}{j!}\binom{j}{k-j}\frac{x^{2j}}{(1-x^2)^{j}}. \end{equation*}