How to show pattern in iterated differentiation

Question

Generally, how would one go about proving general patterns of $n$th derivative?

The specific problem:

$$f(x)=(4-x)^{-0.5}$$

Show that:

$$f^n(0) = 0.5 \left( \frac{(1)(3)(5)\cdots(2n-1)}{8^n} \right)$$

Answer 1

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.