# How would I write the product of something whilst also omitting an element?

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?

• just add $i \neq k$ under the $i=1$ subtext, that's usually how I've seen it May 22, 2020 at 10:43
• Or you could write it as two products: $$\prod_{i=1}^{k-1} x_i^{\alpha_i} \prod_{i=k+1}^n x_i^{\alpha_i}$$ May 22, 2020 at 10:49
• 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 $x_k\partial f/\partial x_k=a_kf$. May 22, 2020 at 11:05
• If I can enjoy the nonzero property, my choice is to write as $\left(\prod x_i^{\alpha_i}\right) / x_k$. May 22, 2020 at 11:05

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.
• @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). May 22, 2020 at 12:31