a[A, 1] = 1; a[A, 2] = 2; a[A, 3] = 3;
a[B, 1] = 5; a[B, 2] = 10; a[B, 3] = 15;

f1[i_, x_] := (1 - x)*Sum[a[i, k]^2, {k, 1, 3}]

Block[{a},Sum[D[f1[A, x], a[i, k]], {i, {A, B}}, {k, {1, 2, 3}}]]
12 (1 - x)

deltaf[f_] := Block[{a}, Sum[D[f, a[i, k]], {i, {A, B}}, {k, {1, 2, 3}}]]

deltaf[f1[A, x]]
0