I am trying to find the second derivative of a product series $\prod_{i=1}^{n} f_i(x) $
I found an answer from Arturo Magidin in First derivative product series, which is pretty convincing and easy to understand.
I tried to follow the same steps to get the second derivative and this is what I got:
$\sum_{i=1}^{n} f_i^{''}(x) \prod_{j=1, j\neq i}^{n} f_j(x)$ + $\sum_{i=1}^{n} f_i^{'}(x) \sum_{j=1, j\neq i}^{n} f_j^{'}(x) \prod_{k=1, k\neq j}^{n} f_k(x)$
My question is, is my derivation correct? If not, where am I making a mistake? And is there any way to simplify this more?
Thanks in advance,