the fyy of -z^2/y
$\displaystyle f(y,z)=\frac{-z^2}{y}$
$\displaystyle f_{yy}(y,z)=\frac{\partial^2f}{\partial y^2}$
All you have to do is treat the z as a constant, and apply the quotient rule successively.
$\displaystyle \frac{\partial}{\partial y}\frac{-(-z^2)}{y^2}=\frac{\partial}{\partial y}\frac{z^2}{y^2}=\frac{-2yz^2}{y^4}=\frac{-2z^2}{y^3}$