It’s a little messy and I don’t understand what you did. But here is how I would do it.
dm/dz= (treat like a constant)* dm/dz(1/z^2), I think you can calculate that
To find dm/dxdz, tit then take dm/dx of that answer.
Treating all non-x-variables as a constant.
Then for dm/dxdz=
-2/z^3 * (cos(xy)/y + x*y(-sin(xy)/y) )