Step 1 :Differentiate the given equation \(\frac{1}{w^{2}+x^{2}}+\frac{1}{w^{2}+y^{2}}=\frac{3}{5}\) implicitly with respect to \(y\).
Step 2 :This gives us \(0 - \frac{2w\frac{\partial w}{\partial y}}{(w^{2}+y^{2})^{2}} = 0\).
Step 3 :Solving for \(\frac{\partial w}{\partial y}\), we find that \(\frac{\partial w}{\partial y} = 0\).
Step 4 :Therefore, the partial derivative of \(w\) with respect to \(y\) at the point \((x, y, w)=(1,3,1)\) is \(\boxed{0}\).