z=x*y+1/x^2+y
z`_(y)=[m]\frac{ ∂ z}{ ∂ y}=(xy+\frac{1}{x^2}+y)`_{y}=x\cdot (y)_{y}`+(x^{-2})_{y}+(y`)_{y}`=x+0+1=x+1[/m]
z``_(xx)=[m]\frac{ ∂^2z}{ ∂x^2}=(y-\frac{2}{x^3})`_{x}=(y)`_{x}-2\cdot (x^{-3})`_{x}=0-2\cdot (-3)\cdot x^{-4}=\frac{6}{x^4}.[/m]
z``_(xy)=[m]\frac{ ∂^2z}{ ∂x ∂y}=(y-\frac{2}{x^3})`_{y}=(y)`_{y}-2\cdot (x^{-3})`_{y}=1[/m]
z``_(yy)=[m]\frac{ ∂^2 z}{ ∂ y^2}=(x+1)`_{y}=0[/m]
z``_(yx)=[m]\frac{ ∂^2z}{ ∂y ∂x }=(x+1)`_{x}=1[/m]
z``_(xy)=z``_(yx)