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.