Derivative of $f(s) = \frac{1}{\prod_{k=1}^{n}(x_k-s)}$

Question

Denote $$ f'_{1}(s) = \bigg( \frac{1}{x_1-s} \bigg)'_{s} = \frac{1}{(x_1-s)^2}\\ f'_{2}(s) = \bigg( \frac{1}{(x_1-s)(x_2 -s)} \bigg)'_{s} = \frac{x_1 +x_2 - 2s}{((x_1-s)(x_2-s))^2} $$

and so on. Is it possible to find a general form of the derivative for $f_n(s) = \frac{1}{\prod_{k=1}^{n}(x_k-s)}$?

I were thinking of something with recurrent expression, but could not come up with anything useful.

This expression arises in characteristic functions of sums of random variables and queueing theory.

Thanks.

Answer 1

Take $\log f(s)$ use implicit differentiation with respect to $s$ on both sides. You will get $$f_{n}^{\prime}(s) = f_{n}(s)\sum_{i=1}^{n}\frac {1}{x_{i}-s}$$.