Find $f_{x}$ and $f_{y}$ for $f(x, y)=y \ln (9 x+y)$.
\[
\begin{array}{l}
f_{x}= \\
f_{y}=
\end{array}
\]
Answer
Thus, the final answer is \(\boxed{f_{x}= \frac{9y}{9x + y}}\) and \(\boxed{f_{y}= \frac{y}{9x + y} + \ln(9x + y)}\).
Step 1 ：Given the function \(f(x, y)=y \ln (9 x+y)\), we need to find the partial derivatives \(f_x\) and \(f_y\).
Step 2 ：To find \(f_x\), we treat \(y\) as a constant and differentiate with respect to \(x\).
Step 3 ：To find \(f_y\), we treat \(x\) as a constant and differentiate with respect to \(y\).
Step 4 ：Applying the rules of differentiation, we find that \(f_x = \frac{9y}{9x + y}\) and \(f_y = \frac{y}{9x + y} + \ln(9x + y)\).
Step 5 ：Thus, the final answer is \(\boxed{f_{x}= \frac{9y}{9x + y}}\) and \(\boxed{f_{y}= \frac{y}{9x + y} + \ln(9x + y)}\).
