Left-hand side:
d(xyz)=xd(yz)+yd(xz)+zd(xy)=
x(ydz+zdy)+y(xdz+zdx)+z(xdy+ydx).
Right-hand side:
cos(x+y+z)(dx+dy+dz).
Now divide through by dx:
x(ydz/dx+zdy/dx)+y(xdz/dx+z)+z(xdy/dx+y)=
cos(x+y+z)(1+dy/dx+dz/dx).
2xydz/dx+2xzdy/dx+2yz-cos(x+y+z)(dy/dx+dz/dx)=cos(x+y+z).
dz/dx(2xy-cos(x+y+z))+dy/dx(2xz-cos(x+y+z)=cos(x+y+z)-2yz.
dz/dx=(cos(x+y+z)-2yz-dy/dx(2xz-cos(x+y+z))/(2xy-cos(x+y+z)).
By symmetry:
dz/dy=(cos(x+y+z)-2xz-dx/dy(2yz-cos(x+y+z))/(2xy-cos(x+y+z)).