OK, though I've done this many times, given that coding this in _Mathematica_ requires one to have a good enough understanding for the core language, let me do a favor:

    Clear[myD, mySum]

    myD[a_ b_, c_] := a myD[b, c] + b myD[a, c]
    (* Add the rule for addition just in case: *)
    myD[a_ + b_, c_] := myD[a, c] + myD[b, c]

    myD[mySum[a_, b__], x_] := mySum[myD[a, x], b]

    myD[(h_ /; Context[h] =!= "System`")[a__], x_] := 
     With[{l = Length@{a}}, 
      Total@MapThread[
        Derivative[Sequence @@ #][h][a] myD[#2, x] &, {IdentityMatrix@
          l, {a}}]]

    myD[Subscript[a_, i___, j1_, k___], Subscript[a_, i___, j2_, k___]] :=
      KroneckerDelta[j1, j2]

    myD[Subscript[a_, __], Subscript[b_, __]] := 0

    mySum[KroneckerDelta@OrderlessPatternSequence[i_, j_] expr_, jL___, 
      j_, jR___] := mySum[expr /. j -> i, jL, jR]

    mySum[a_] := a

    MakeBoxes[mySum[a__], fmt_] ^:= 
     With[{funcbody = 
        MakeBoxes[Sum[a], fmt] /. Thread[{a} -> (Slot /@ Length@{a})]}, 
      TemplateBox[MakeBoxes[#, fmt] & /@ {a}, "mySum", 
       DisplayFunction -> (funcbody &)]]

Usage:

    myD[mySum[Subscript[a, i, j] Subscript[x, j], j], Subscript[a, i, k]]

[![enter image description here][1]][1]

    myD[Subscript[x, k] mySum[g[Subscript[a, i, j], 
                                Subscript[x, j]], i, j], Subscript[x, n]]

[![enter image description here][2]][2]

You can press <kbd>Ctrl</kbd>+<kbd>Shift</kbd>+<kbd>t</kbd> to transform the output to `TraditionalForm`, and <kbd>Ctrl</kbd>+<kbd>Shift</kbd>+<kbd>n</kbd> to transform the output to `StandardForm`.

  [1]: https://i.sstatic.net/OO6L5D18.png
  [2]: https://i.sstatic.net/cWzUKBng.png