why index notation of summation changes during derivative in chain rule?

Question

I am trying to understand a derivative example from a class note. As shown in the picture, it's working out the derivative of a $log(exp())$ function with respect to $V_c$. When applying the chain rule, the index of summation $\sum\limits_{w=1}^V$ changed from $w$ to $x$ and the note says "Important to change index". I'm a bit confused why we have to change the index. Thanks. Picture to the example

Answer 1

It's not apparent from that image alone why it should be important to change the index. My guess would be that they later go on to write this as

$$\sum_{x=1}^Va_xu_x$$

with weights

$$ a_x=\frac{\exp\left(u_x^\top V_c\right)}{\sum_{w=1}^V\exp\left(u_w^\top V_c\right)}\;, $$

and in that case, when the index is no longer just a bound summation index but appears as a free variable, it's important not to use the same letter for it as for a summation index in the same expression. As long as you just have two summations in the same expression and they're not nested, it's not a problem if they use the same summation index.