Find \( \frac{d y}{d x} \) by implicit differentiation. \[ e^{x / y}=5 x-y \] \( \frac{d y}{d x}=\square \)

To find \( \frac{d y}{d x} \) using implicit differentiation, we start with the equation: \[ e^{x/y} = 5x - y \] Now, differentiate both sides with respect to \( x \). Using the chain rule on the left side, we get: \[ \frac{d}{dx}\left(e^{x/y}\right) = e^{x/y} \cdot \left(\frac{d}{dx}\left(\frac{x}{y}\right)\right) \] For \(\frac{d}{dx}\left(\frac{x}{y}\right)\) we can apply the quotient rule: \[ \frac{d}{dx}\left(\frac{x}{y}\right) = \frac{y \cdot 1 - x \cdot \frac{dy}{dx}}{y^2} = \frac{y - x\frac{dy}{dx}}{y^2} \] So, we have: \[ \frac{d}{dx}\left(e^{x/y}\right) = e^{x/y} \cdot \frac{y - x\frac{dy}{dx}}{y^2} \] On the right side, we differentiate: \[ \frac{d}{dx}(5x - y) = 5 - \frac{dy}{dx} \] Now, we put it all together: \[ e^{x/y} \cdot \frac{y - x\frac{dy}{dx}}{y^2} = 5 - \frac{dy}{dx} \] Next, we multiply through by \( y^2 \): \[ e^{x/y} (y - x\frac{dy}{dx}) = y^2(5 - \frac{dy}{dx}) \] Distributing gives: \[ e^{x/y} y - e^{x/y} x \frac{dy}{dx} = 5y^2 - y^2 \frac{dy}{dx} \] Now, let's gather all terms involving \( \frac{dy}{dx} \) on one side: \[ e^{x/y} y - 5y^2 = e^{x/y} x \frac{dy}{dx} - y^2 \frac{dy}{dx} \] Factoring out \( \frac{dy}{dx} \) on the right side yields: \[ e^{x/y} y - 5y^2 = \left(e^{x/y} x - y^2\right) \frac{dy}{dx} \] Hence, \[ \frac{dy}{dx} = \frac{e^{x/y} y - 5y^2}{e^{x/y} x - y^2} \] So, the final answer is: \[ \frac{dy}{dx} = \frac{e^{x/y} y - 5y^2}{e^{x/y} x - y^2} \]