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
$o$.