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I have a function $$f(x_1,x_2,...,x_n) = \prod_{i=1}^{n}x_i^{\alpha_{i}}$$

I want to take the partial derivative $$\frac{\partial f}{\partial x_k}$$

Now I believe this will look like $${\alpha_kx^{\alpha_k-1}} \cdot \prod_{i=1}^{n} x_i^{\alpha_i} $$

This isn't actually correct. The last product will be omitting the $k^{th}$ term. But I am not sure how to write that out?

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    $\begingroup$ just add $i \neq k$ under the $i=1$ subtext, that's usually how I've seen it $\endgroup$ May 22, 2020 at 10:43
  • $\begingroup$ Or you could write it as two products: $$\prod_{i=1}^{k-1} x_i^{\alpha_i} \prod_{i=k+1}^n x_i^{\alpha_i} $$ $\endgroup$
    – Vishu
    May 22, 2020 at 10:49
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    $\begingroup$ If you aren't concerned with dividing by $0$, you can write it as $(a_k/x_k)f$. To be a bit more careful, you can write $x_k\partial f/\partial x_k=a_kf$. $\endgroup$ May 22, 2020 at 11:05
  • $\begingroup$ If I can enjoy the nonzero property, my choice is to write as $\left(\prod x_i^{\alpha_i}\right) / x_k$. $\endgroup$
    – dust05
    May 22, 2020 at 11:05

1 Answer 1

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The standard notation is to use a hat: writing, for instance, $(a_1,...\widehat{a}_i, …, a_n)$ means that you do not consider the $i$-th term. In this case, however, simply writing: $$ \prod_{1 \leq i \leq n, \, \, i \neq k} x_i ^{\alpha_i} $$ works as well.

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    $\begingroup$ That's it, +1. However I think that the "hat notation" is not THAT standard and sometimes it can be tricky. If possible, better to avoid it. $\endgroup$ May 22, 2020 at 11:02
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    $\begingroup$ @GiuseppeNegro. The hat notation is very common and can be considered standard. $\endgroup$
    – md2perpe
    May 22, 2020 at 12:06
  • $\begingroup$ @md2perpe: Let me elaborate. The best notation is the one that is less encumbering and that requires less explanation. In this case, writing $$\prod_{1\le i \le n,\ i\ne j}, $$ or even $$\prod_{i\ne j}$$ will immediately be clear to everybody with a minimal mathematical background. On the other hand, the hat notation $$\prod_{1\le i \le n} x_i^{\alpha_i}\widehat{x_j^{\alpha_j}}$$ is both more cumbersome and requires at least a couple of words of explanation. Besides, the hat is widely used and it can be mistaken with something else (the Fourier transform, for example). $\endgroup$ May 22, 2020 at 12:31
  • $\begingroup$ @GiuseppeNegro. This is the first time I see it used with the product symbol, I think. In that context it doesn't work very well. $\endgroup$
    – md2perpe
    May 22, 2020 at 12:58

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