The Antialiasing option of Style only affects 2D graphics. For 3D plots the setting is global. To alter it, in the Mathematica Notebook menu choose Edit→Preferences, in dialog that appears go to Appearance tab, there to Graphics nested tab. Then move the slider to choose the amount of antialiasing.
If it's just for display, then I think the following is enough:
myD[myD[a_, b__], c__] := myD[a, b, c]
Format[HoldPattern@myD[a_, b__]] := TraditionalForm@HoldForm@D[a, b]
Format[x[a_, b_]] := Subscript[x, a, b]
myD[myD[S, x[ν, μ]], x[α, β]]
To automate the subsequent calculation, just define
myD[a_ + b_, c_] := myD[a, c] + myD[b, c]