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What I am interested in is

$$ \frac{d}{dx}\prod_{a=1}^{b}f_{a}(x). $$ I know that a derivative can easily be distributed into a summation, but what about an arbitrary product?

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1 Answer 1

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By using product rule, we have

$$\frac{d}{dx}\prod_{a=1}^b f_a(x) = \sum_{a=1}^b \left(\frac{d}{dx} f_a(x) \right)\prod_{j \ne a}f_j(x)$$

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  • $\begingroup$ If this is an infinite product, for which functions can we do this and what are assumptions which make the right hand side convergent? Just for curiosity. $\endgroup$
    – Jonas Lenz
    Oct 1, 2018 at 5:16
  • $\begingroup$ oh, I am assuming finite product. I am not sure about infinite product. $\endgroup$ Oct 1, 2018 at 5:18
  • $\begingroup$ Do you mind proving that? There are ways I can play around with a product and sum to reasonably assume that form from my own interpretation, but I have never seen anyone actually prove how the product rule works for any number of factors, only 2 or 3. $\endgroup$
    – DaneJoe
    Oct 1, 2018 at 5:18
  • $\begingroup$ try using mathematical induction, treat $k$ product as a term as an intermediate step. $\endgroup$ Oct 1, 2018 at 5:21
  • $\begingroup$ So what is the j here? Does it count at the same time as a? Is it part of the sum? $\endgroup$
    – DaneJoe
    Oct 1, 2018 at 5:53

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