Find $f_{x}$ and $f_{y}$ for $f (x, y) = \frac{4 x}{9 y} - \frac{2 y}{5 x}$
$\begin{array}{l} f_{x} = \\ f_{y} = \end{array}$

Answer

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Answer

$f_{y} = - \frac{4 x}{9 y^{2}} - \frac{2}{5 x}$

Steps

Step 1 ：Given the function $f (x, y) = \frac{4 x}{9 y} - \frac{2 y}{5 x}$ , 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 ：The partial derivative $f_{x}$ is calculated as $f_{x} = \frac{4}{9 y} + \frac{2 y}{5 x^{2}}$ .

Step 5 ：The partial derivative $f_{y}$ is calculated as $f_{y} = - \frac{4 x}{9 y^{2}} - \frac{2}{5 x}$ .

Step 6 ：Thus, the final answers are $f_{x} = \frac{4}{9 y} + \frac{2 y}{5 x^{2}}$ and $f_{y} = - \frac{4 x}{9 y^{2}} - \frac{2}{5 x}$ .

Step 7 ： $f_{x} = \frac{4}{9 y} + \frac{2 y}{5 x^{2}}$

Step 8 ： $f_{y} = - \frac{4 x}{9 y^{2}} - \frac{2}{5 x}$