If $y=\sum_i\frac{x_i}{\sigma_i^2}$ then is $\frac{\partial y}{\partial x_i}=\sum_i\frac{1}{\sigma_i^2}$ or is it equal to $\frac{1}{\sigma_i^2}$?

Question

I thought that the former is correct, but then I read this as part of certain proof:

Should the sigma be omitted on taking the partial derivative?

Answer 1

Let me give an example and hopefully things clear up:

$$f=x+2y$$

$$\frac{\partial f}{\partial x}=1$$ $$\frac{\partial f}{\partial y}=2$$