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