\partial_{y} f(0,0)=\lim_{t \rightarrow 0}\frac{f(0,t)-f(0,0)}{t}=
\lim_{t \rightarrow 0}\frac{\sqrt{2t^4}}{t}=\lim_{t \rightarrow 0}\frac{\sqrt{2}t^2}{t}=\lim_{t \rightarrow 0}\sqrt{2}t=0