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 $o$.