Define an $n$-variable function with $n$ undefined?

You can do this:

sum = Sum[i*Indexed[x, i], {i, 1, n}]

D[sum, Indexed[x, 5]]
(* Piecewise[{{5, n >= 5}}, 0] *)

If you have your function take in a list, so you can deal with arbitrary length inputs...

ff[lst_?ListQ] := Total@lst

Then you can do

xs = Array[x, 5]
(* {x[1], x[2], x[3], x[4], x[5]} *)

D[ff[xs], x[4]]
(* 1 *)

D[ff[xs], x[40]]
(* 0 *)

Slightly more complicated

gg[lst_?ListQ] := Times @@ lst

D[gg[xs], x[1]]
(* x[2] x[3] x[4] x[5] *)