Answer
$\frac{\partial^{2} s}{\partial x^{2}}$ = $\frac{2xy}{(x^{2}+y^{2})^{2}}$
$\frac{\partial^{2} s}{\partial x{\partial y}}$ = $\frac{-x^{2}+y^{2}}{(x^{2}+y^{2})^{2}}$
$\frac{\partial^{2} s}{\partial y^{2}}$ = $\frac{-2xy}{(x^{2}+y^{2})^{2}}$
Work Step by Step
$\frac{\partial s}{\partial x}$ = $\frac{1}{1+(\frac{y}{x})^{2}}$$[\frac{-y}{x^{2}}]$ = $\frac{x^{2}}{x^{2}+y^{2}}$$[\frac{-y}{x^{2}}]$ = $\frac{-y}{x^{2}+y^{2}}$
$\frac{\partial^{2} s}{\partial x^{2}}$ = $\frac{-(-y)(2x)}{(x^{2}+y^{2})^{2}}$ = $\frac{2xy}{(x^{2}+y^{2})^{2}}$
$\frac{\partial^{2} s}{\partial x{\partial y}}$ = $\frac{(x^{2}+y^{2})(-1)-(-y)(2y)}{(x^{2}+y^{2})^{2}}$ = $\frac{-x^{2}-y^{2}+2y^{2}}{(x^{2}+y^{2})^{2}}$ = $\frac{-x^{2}+y^{2}}{(x^{2}+y^{2})^{2}}$
$\frac{\partial s}{\partial y}$ = $\frac{1}{1+(\frac{y}{x})^{2}}$$[\frac{x}{x^{2}}]$ = $\frac{x^{2}}{x^{2}+y^{2}}$$[\frac{1}{x}]$ = $\frac{x}{x^{2}+y^{2}}$
$\frac{\partial^{2} s}{\partial y^{2}}$ = $\frac{(-x)(2y)}{(x^{2}+y^{2})^{2}}$ = $\frac{-2xy}{(x^{2}+y^{2})^{2}}$
