Partial differentiation second order

Question

I have some rules for differentiation:
myD[-a_, o_] := -myD[a, o];
myD[a_, -o_] := -myD[a, o];
myD[a_ + n_, o_] := myD[a, o] + myD[n, o];
myD[a_ b_, o_] := b myD[a, o] + a myD[b, o];
myD[myD[a_, b__], c__] := myD[a, b, c]
Format[HoldPattern@myD[a_, b__]] := TraditionalForm@HoldForm@D[a, b]
$Assumptions = 
 x [Element] Matrices[{2*M, 2*M}, Reals, Antisymmetric[{1, 2}]]
x[arg__] /; ! OrderedQ@{arg} := Signature@{arg} x @@ Sort@{arg} 
Format[x[arg__]] := Subscript[x, arg]

What to do to create a new rule that defines,
$$frac{partial ^2Q}{partial x_{1,2}, partial x_{3,4}}=frac{partial ^2Q}{partial x_{3,4}, partial x_{1,2}}$$
myD[myD[Q, x[1, 2]], x[3, 4]]

is how to write code to get $frac{partial ^2Q}{partial x_{1,2}, partial x_{3,4}}$

Lukas Lang · Accepted Answer

You can do something like this:
myD[a_, o__] /; ! OrderedQ@{o} := myD[a, Sequence @@ Sort@{o}]

myD[myD[Q, x[1, 2]], x[3, 4]]

This will ensure that the coordinates after which the expression is differentiated are always sorted. You can define another order instead of Sort{o}, but the key is that all equivalent forms are converted into a single canonical version:
myD[myD[Q, x[1, 2]], x[3, 4]] == myD[myD[Q, x[3, 4]], x[1, 2]] 
(* True *)

Partial differentiation second order

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