\partial^2_{y} f(0,0)=\lim_{t \rightarrow 0}\frac{\partial_{y}f(0,t)-\partial_{y}f(0,0)}{t}=\lim_{t \rightarrow 0}\frac{\frac{8t^3}{\sqrt{2t^4 }}-0}{t}=\lim_{t \rightarrow 0} 2^{\frac{5}{2}}=2^{\frac{5}{2}}