设二阶偏导数连续的函数z=f(u,v),u=xy,v=x^2-y^2,求δz/δx,δz/δy,最好有详细的步骤

z=f(u,v),u=xy,v=x^2-y^2
du/dx=y,du/dy=x
dv/dx=2x,dv/dy=-2y
dz/dx=dz/du*du/dx+dz/dv*dv/dx
=df/du*y+df/dv*2x
dz/dy=dz/du*du/dy+dz/dv*dv/dy
=df/du*x-df/dv*2y