Define a derivative-like function

Question

I would like to define a function Cd[a_][expr_] which takes an arbitrary argument a and acts on the expression expr by following a chain rule, in a way that resembles derivatives.

Let's consider two examples.

If expr = p G[x], for some constant p, I would like to get Cd[a][expr] = Cd[a][p] G[x]+p Cd[a][G[x]]

If expr = I p G[x], for some constant p, I would like to get Cd[a][expr] = Cd[a][I] p G[x]+I Cd[a][p] G[x]+I p Cd[a][G[x]]

One possible definition of this function is

Cd[a_][expr_?((Head[#] == Times) &)] := Module[{listexpr}, listexpr = List @@ expr;
Table[Cd[a][listexpr[[i]]] Times @@ DeleteCases[listexpr, listexpr[[i]]], 
{i, 1,Length[listexpr]}] // Total // Return]
Cd[a_][b_?NumberQ] = 0;

However, I wonder whether this definition can be simplified. Any suggestion? Thanks!

Answer 1

A recursive definition could work:

Cd[a_][u_*v_] := Cd[a][u]*v + u*Cd[a][v]

Cd[a][p*G[x]]
(*    G[x]*Cd[a][p] + p*Cd[a][G[x]]    *)

Cd[a][I*p*G[x]]
(*    p*G[x]*Cd[a][I] + I*(G[x]*Cd[a][p] + p*Cd[a][G[x]])    *)

Keep in mind that ratios of the form u/v == Times[u, 1/v] also have the head Times:

Cd[a][u/v]
(*    Cd[a][u]/v + u Cd[a][1/v]    *)