<s>[INST] How to show pattern in iterated differentiation Generally, how would one go about proving general patterns of $n$th derivative?
The specific problem:
$$f(x)=(4-x)^{-0.5}$$
Show that:
$$f^n(0) = 0.5 \left( \frac{(1)(3)(5)\cdots(2n-1)}{8^n} \right)$$
 [/INST] We can write this formally (using the odd factorial)
$$ f^{(n)}(0) = \frac{1}{2^{3n+1}} \prod_{k=1}^n (2k-1) = \frac{1}{2^{4n+1}}  \frac{(2n)!}{n!}
$$
Using the power rule,
$$ f^{(n)}(x) = (-1)^n(4-x)^{-1/2 - n} \prod_{k=1}^{n} \left(\frac{1}{2} - k\right) = (4-x)^{-1/2 - n} \frac{1}{2^n} \prod_{k=1}^{n} \left(2k - 1\right)
$$
Let $x = 0$, we have
$$ (4-0)^{-1/2 - n} \frac{1}{2^n} \prod_{k=1}^{n} \left(2k - 1\right)= \frac{1}{2^{2n+1}} \frac{1}{2^n} \frac{(2n)!}{2^n n!} = \frac{1}{2^{4n+1}}  \frac{(2n)!}{n!}
$$
as desired.
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