X = {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]};
r := Sqrt[Sum[(X[[i]]^2, {i, 3}]];
T[i_, j_] :=KroneckerDelta[i, j]/r + X[[i]] X[[j]] / r^3
TD[i_, j_, k_] := D[GF[i, j], X[[k]]];

TD[i, j, k]
0

X = Array[Subscript[x, #] &, 3];
T = Outer[Times, X, X];

TD = D[T, {X}];
TD2 = Array[KroneckerDelta[#, #3] Subscript[x, #2] + Subscript[x, #] KroneckerDelta[##2] &, {3, 3, 3}];

TD == TD2

True