du/dy
=1/(dy/du)
=(1+uv)^2
dv/dy
=1/(dy/dv)
=-(1+uv)^2/u^2
dz/dx
=dz/du*du/dx+dz/dw*dw/du*du/dx
=[(1+uw)-u(w+udw/du)]/(1+uw)^2*1-u^2/(1+uw)^2*dw/du*1
=1/(1+uw)^2
dz/dy
=dz/du*du/dy+dz/dw*(dw/du*du/dy+dw/dv*dv/dy)
=[(1+uw)-u(w+udw/du)]/(1+uw)^2*(1+uv)^2
-u^2/(1+uw)^2*((1+uv)^2*dw/du-(1+uv)^2/u^2*dw/dv)
x^2*dz/dx+y^2*dz/dy
=x^2*[1/(1+uw)^2]+y^2*{[(1+uw)-u(w+udw/du)]/(1+uw)^2*(1+uv)^2-u^2/(1+uw)^2*((1+uv)^2*dw/du-(1+uv)^2/u^2*dw/dv)}
=u^2*[1/(1+uw)^2]+[u/(1+uv)^2]^2*{[(1+uw)-u(w+udw/du)]/(1+uw)^2*(1+uv)^2-u^2/(1+uw)^2*((1+uv)^2*dw/du-(1+uv)^2/u^2*dw/dv)}=...=u^2/(1+uw)^2
=>(1-u^2*dw/du)=u^2*dw/du-dw/dv
=>2u^2*dw/du-dw/dv=1