Find $f_{x}$ and $f_{y}$ for $f(x, y)=\frac{4 x}{9 y}-\frac{2 y}{5 x}$ \[ \begin{array}{l} f_{x}= \\ f_{y}= \end{array} \]
Solution
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 \(f_x = \frac{4}{9y} + \frac{2y}{5x^{2}}\).
Step 5 ：The partial derivative \(f_y\) is calculated as \(f_y = -\frac{4x}{9y^{2}} - \frac{2}{5x}\).
Step 6 ：Thus, the final answers are \(f_x = \frac{4}{9y} + \frac{2y}{5x^{2}}\) and \(f_y = -\frac{4x}{9y^{2}} - \frac{2}{5x}\).
Step 7 ：\(\boxed{f_{x}=\frac{4}{9y} + \frac{2y}{5x^{2}}}\)
Step 8 ：\(\boxed{f_{y}=-\frac{4x}{9y^{2}} - \frac{2}{5x}}\)
