Find $f_{x}$ and $f_{y}$ for $f (x, y) = y \ln (9 x + y)$ .
$\begin{array}{l} f_{x} = \\ f_{y} = \end{array}$

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Answer

Thus, the final answer is $f_{x} = \frac{9 y}{9 x + y}$ and $f_{y} = \frac{y}{9 x + y} + \ln (9 x + y)$ .

Steps

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 $f_{x} = \frac{9 y}{9 x + y}$ and $f_{y} = \frac{y}{9 x + y} + \ln (9 x + y)$ .

Step 5 ：Thus, the final answer is $f_{x} = \frac{9 y}{9 x + y}$ and $f_{y} = \frac{y}{9 x + y} + \ln (9 x + y)$ .