z = sin(2xy) Ill use D for the partial differential sign
Dz/Dx = 2ycos(2xy)
Dz/Dy = 2xcos(2xy)
D^2z/Dx^2 = 2y.-2ysin(2xy) = - 4y^2sin(2xy)
D^2z/Dy^2 = -4x^2sin(2xy)
D^2z/Dy.Dx = D/Dy( Dz/Dx) = 2y. -2xsin(2xy) + 2.cos(2xy)
= -4xysin(2xy) + 2 cos(2xy)
D^2z(Dx.Dy) is the same as the latter.