\[ \begin{aligned} \frac{\partial f}{\partial x} &= \frac{\partial}{\partial x} \left( \frac{x}{y} \right) = \frac{\partial}{\partial x} \left( x \cdot \frac{1}{y} \right) = \frac{1}{y} \frac{\partial}{\partial x}(x) + x \frac{\partial}{\partial x}\left( \frac{1}{y} \right) \\ &= \frac{1}{y} \cdot 1 + x \cdot 0 = \frac{1}{y} \end{aligned} \]
\[ \begin{aligned} \frac{\partial f}{\partial y} &= \frac{\partial}{\partial y} \left( \frac{x}{y} \right) = \frac{\partial}{\partial y} \left( x \cdot \frac{1}{y} \right) = \frac{1}{y} \frac{\partial}{\partial y}(x) + x \frac{\partial}{\partial y}\left( \frac{1}{y} \right) \\ &= \frac{1}{y} \cdot 0 + x \left( -\frac{1}{y^2} \right) = -\frac{x}{y^2} \end{aligned} \]