Answer
$f_{x}=\frac{x-y}{x^{2}+y^{2}}$
$f_{y}=\frac{x+y}{x^{2}+y^{2}}$
Work Step by Step
$f(x,y)=\frac{1}{2}\ln(x^{2}+y^{2})+tan^{-1}\frac{y}{x}$
$f_{x}=\frac{1}{2}\frac{2x}{x^{2}+y^{2}}+\frac{-\frac{y}{x^2}}{1+(\frac{y}{x})^2}=\frac{x}{x^{2}+y^{2}}-\frac{y}{x^{2}+y^{2}}=\frac{x-y}{x^{2}+y^{2}}$
$f_{y}=\frac{1}{2}\frac{2y}{x^{2}+y^{2}}+\frac{\frac{1}{x}}{1+(\frac{y}{x})^2}=\frac{x+y}{x^{2}+y^{2}}$
