Defining a derivative on a custom function

Question

I am trying to define a derivative that acts on a custom function, $f$, such that when it acts, it increments one of the arguments of by 1. f is not defined; I just want to use it to keep track of derivatives. I know I could define a separate operator to do this, but this function, $f$, will be multiplied by other functions that need differentiating, so I thought it easier to define how derivatives act on f. My attempt thus far:

Derivative[1, 0, 0, 0][f[a,b,c,d]][x1, x2, x3, x4] = f[a,b,c,d]

However, when I go to evaluate this derivative, I get

Input: D[f[a,b,c,d][x1, x2, x3, x4], x1]
Output: f[a,b,c,d]^{1,0,0,0}[x1, x2, x3, x4]

whereas, I would hope to get something like:

Input: D[f[a,b,c,d][x1, x2, x3, x4], x1]
Output: f[a+1,b,c,d][x1, x2, x3, x4]

The derivative definition only works specifically for the exact syntax in the definition, and not with any other arguments apart from a,b,c,d:

Input: D[f[a,b,c,d][x1, x2, x3, x4], x1]
Output: f[a+1,b,c,d]

Any suggestions on how to obtain the second listed output?

Answer 1

It seems to be that all you need is SetDelayed (:=) instead of Set (=):

Derivative[1, 0, 0, 0][f][a_, b_, c_, d_] := f[a + 1, b, c, d]

Now let us evaluate some derivatives:

D[f[x, y, z, p], {x, 1}] (* with respect to that first argument *)
D[f[x, y, z, p], {y, 1}] (* with respect to any other argument *)

(*Out: 
f[1 + x, y, z, p]
Derivative[0, 1, 0, 0][f][x, y, z, p]
*)