Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 4 down vote accepted

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

share|improve this answer
    
-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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.