Sums with arguments x_1...x_infinity

Questions
1

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,x_2] 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: Am I allowed to use subscript in function arguments? , 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.

2

Let t\in \mathbb{Q} to stay in the countable area. Then you consider x[t] as variable different from x[t'] for t\not=t'? With other words, one cannot consider derivatives of x[t] with respect to t because t is an index? This can be evaluated

 In[8]:= Clear[f]
    f[x_[t_], y_[t_]] := Sin[x[t]^2 + y[t]^3] + Cos[x[t]^3 - y[t]^2]
    
 In[10]:= D[f[p[o], q[o]], p[o]]
 Out[10]= 2 Cos[p[o]^2 + q[o]^3] p[o] - 3 p[o]^2 Sin[p[o]^3 - q[o]^2]

but not

In[11]:= D[f[p[o], q[o]], p[2 o]]
Out[11]= 0

In[12]:= D[f[p[o], q[o]], p[o + 1]]
Out[12]= 0

in the last two cases - I presume - you expect to get a part of G differentiated where the variable p[2 o] etc. appears once the o is fixed. That you can only reach at - irony - if again p[\xi[o]] is related to p[o] - with other words, p[\xi[o]] and p[o] are not independent variables, but only o.