\frac{\partial ^2r}{\partial x\partial y}=\frac{d}{\text{dy}}\left(\frac{\partial r}{\partial x}\right)=\frac{d}{\text{dy}}\left(\frac{2x}{z^3} f'\left(\frac{x^2}{y z^3}\right)\right)=\frac{2x}{z^3}f\text{''}\left(\frac{x^2}{y z^3}\right)\text{  }\frac{d}{\text{dy}}\left(\frac{x^2}{y z^3}\right)=\frac{2x}{z^3}f\text{''}\left(\frac{x^2}{y z^3}\right)\frac{x^2}{z^3}\left(\frac{-1}{y^2}\right)=-2\frac{x^3}{y^2z^6}f\text{''}\left(\frac{x^2}{y z^3}\right)