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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.


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up vote 4 down vote accepted

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

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-1. Obviously false already for $n=1$... – Hans Lundmark Sep 12 '12 at 8:35
sorry ! typo fixed. – s.b Sep 12 '12 at 12:47
Downvote removed. – Hans Lundmark Sep 12 '12 at 14:30

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