Differentiating with respect to size of index

Question

I have the following function: $$a\sum_{i=1}^{n}x_i. $$ I wish to differentiate with respect to $n$. If all $x_i$s were the same, this problem would be trivial, obviously. Can anyone help?

Answer 1

Depending on how deep you go into integration theory, your question may not make a lot of sense.

Notice that you are differentiating with respect to a non continous variable ($n \notin \mathbb R $) but to a discrete variable ($n \in \mathbb N $).

To go further, making sense to this differential would involve distribution theory wikipedia : Distribution

Answer 2

I suppose you could turn it into a continuous variable as follows: $$\int_{0}^{\overline{z}}z dz $$. differentiating this expression wrt $\overline{z}$ yields $\overline{z}$?