Derivative of a function in undefined dimension

Question

I have a scalar function defined on a $ n $-dimensional manifold: $ f(x_1, x_2, ..., x_n) $, where $ n $ is undefined, and $x_i$ are the coordinates. How to define something like "$∂_af∂^af$"?

(I'm solving the Einstein equation for a black brane in the large-N limit, where N is the dimension of the brane, so I should keep N in my expression instead of setting N as something like 5)

I've tried:

In[10]:=f/:D[f[i_],x[j_]]=f[i+x[j]]
In[11]:=D[f[0],x[5]]
Out[11]:=f[x[5]]

That's OK but then

In[13]:=D[-f[0],x[5]]
Out[13]:=0

It doesn't work now:(

Edit:

My current solution is just

SetOptions[D, NonConstants -> {f}]

It almost perfectly solved my problem despite the complicated output. I'm not trying to simplify the output.

Answer 1

If only constants are the problem, this could help:

 x /: D[c_. f[i_], x[j_]] = c f[i + x[j]]