A real-valued function $ f$ satisfies for all reals $ x$ and $ y$ the equality \[ f (xy) = f (x)y + x f (y). \] Prove that this function satisfies for all reals $ x$ and $ y \ne 0$ the equality \[ f\left(\frac{x}{y}\right)=\frac{f (x)y - x f (y)}{y^2} \]