Calculate the partial derivative $\frac{\partial w}{\partial y}$ using implicit differentiation of $\frac{1}{w^{2}+x^{2}}+\frac{1}{w^{2}+y^{2}}=\frac{3}{5}$ at $(x, y, w)=(1,3,1)$.
(Use symbolic notation and fractions where needed.)
\[
\frac{\partial w}{\partial y}=
\]
Answer
Therefore, the partial derivative of \(w\) with respect to \(y\) at the point \((x, y, w)=(1,3,1)\) is \(\boxed{0}\).
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 \(\boxed{0}\).
