Dear community, I am trying to define and symbolically differentiate a function of the following form: $$G=\sum_{x_t,y_t=1}^T f(x_t,y_t) $$ Where $t$ denotes a certain period of time and $f (x_t, y_t) $ is some function such as $aln(x_t)+bln(y_t) $. Is there any way to implement this so that something like D[G,x2] or D[G,x(t+1)] would produce meaningful results?

Note: I've tried to use the symbolize command as described by Bruce Miller in this thread: http://community.wolfram.com/groups/-/m/t/272608, but this only works for a finite number of arguments $x_1$ through $x_t$; I wish to implement this for a function with countably many x's.