How to make a matrix that had diffrentation without functions

One possiblity is to define the function d as:

d = {{D[#, x], 0}, {0, D[#, y]}, {D[#, x], D[#, y]}} &

To apply this operator to a particular function:

f[x_, y_] := x^2 + Sin[y]

Then:

d[f[x, y]]

{{2 x, 0}, {0, Cos[y]}, {2 x, Cos[y]}}