Unexpected behavior with Derivative and a function parameters defined via an Association

Question

I am confused by the following behavior:

ClearAll[u, i, x, y, func, z];
u = <|1 -> 1, 2 -> 2|>;
func[x_, i_, y_] := x u[i] + y u[i] + z[i][y];
Derivative[0, 0, 1][func][x, 2, y]

I expect the derivative to be u[2]+z[2]'[y] and since u[2] is 2 here that should be 2 + z[2]'[y]

Why is the result

Missing["KeyAbsent", 2] + Derivative[1][z[2]][y]

The final term makes perfect sense, but why the "KeyAbsent" message?

Answer 1

Why is the result

Missing["KeyAbsent", 2] + Derivative[1][z[2]][y]

Heads are evaluated first. The head of Derivative[0, 0, 1][func][x, 2, y] is Derivative[0, 0, 1][func], which yields the following:

Derivative[0, 0, 1][func]
(*  Missing["KeyAbsent", #2] + Derivative[1][z[#2]][#3] &  *)

This shouldn't be completely surprising because argument i has not yet been given a value. The way Mathematica computes Derivative[0, 0, 1][func] is to compute the derivative of func at private dummy variables, which I'll abbreviate as follows:

func[X[1], X[2], X[3]]
(*
Missing["KeyAbsent", X[2]] X[1] + Missing["KeyAbsent", X[2]] X[3] + 
 z[X[2]][X[3]]
*)

This is differentiated with respect to X[3] in this case. Then the dummy variables X[..] are replaced with Slot[..] (#2 etc.) as shown above. Finally the arguments x, 2, y are injected into the correspond slots.

The problem that Mathematica is addressing is that the derivative with respect to some variable(s) cannot be computed if you first replace the variables by some values. Clearest example: In Derivative[1,2,3][f][5,7,9], we cannot find the derivative of f after we plug in the constants 5,7,9.

Hence @Nasser's solution is the way to go.

Another approach is to use subvalues to protect nondifferentiable parameters from being differentiated:

ClearAll[u, i, x, y, func, z];
u = <|1 -> 1, 2 -> 2|>;
func[i_Integer][x_, y_] := x u[i] + y u[i] + z[i][y];
Derivative[0, 1][func[2]][x, y]
(*  2 + Derivative[1][z[2]][y]  *)