Find $f_{x}$ and $f_{y}$ for $f(x, y)=y \ln (3 x+8 y)$.

Step 1 ：Given the function \(f(x, y)=y \ln (3 x+8 y)\), we are asked to find the partial derivatives \(f_x\) and \(f_y\).

Step 2 ：To find \(f_x\), we differentiate \(f(x, y)\) with respect to \(x\) while treating \(y\) as a constant.

Step 3 ：To find \(f_y\), we differentiate \(f(x, y)\) with respect to \(y\) while treating \(x\) as a constant.

Step 4 ：We use the rule that the derivative of \(\ln(u)\) is \(\frac{1}{u}\) times the derivative of \(u\).

Step 5 ：We also use the product rule for differentiation, which states that the derivative of \(u \cdot v\) is \(u' \cdot v + u \cdot v'\), where \(u'\) and \(v'\) are the derivatives of \(u\) and \(v\) respectively.

Step 6 ：Applying these rules, we find that \(f_x = \frac{3y}{3x+8y}\).

Step 7 ：Similarly, we find that \(f_y = \frac{8y}{3x+8y} + \ln(3x+8y)\).

Step 8 ：Thus, the partial derivatives of the function \(f(x, y)=y \ln (3 x+8 y)\) are \(\boxed{f_{x} = \frac{3y}{3x+8y}}\) and \(\boxed{f_{y} = \frac{8y}{3x+8y} + \ln(3x+8y)}\).