Calculate the derivative of \( y \) with respect to \( x \) \( x e^{y}=5 x y+2 y^{3} \) \( \frac{d y}{d x}=\square \)

To find the derivative of \( y \) with respect to \( x \), we will differentiate both sides of the equation \( x e^{y} = 5xy + 2y^{3} \) using implicit differentiation. Differentiating the left side, we have: \[ \frac{d}{dx}(x e^{y}) = e^{y} + x e^{y} \frac{dy}{dx} \] For the right side: \[ \frac{d}{dx}(5xy + 2y^{3}) = 5\left(y + x\frac{dy}{dx}\right) + 6y^{2}\frac{dy}{dx} \] Setting both derivatives equal gives us: \[ e^{y} + x e^{y} \frac{dy}{dx} = 5\left(y + x\frac{dy}{dx}\right) + 6y^{2}\frac{dy}{dx} \] Now, we can isolate \( \frac{dy}{dx} \): \[ x e^{y} \frac{dy}{dx} - 5x \frac{dy}{dx} - 6y^{2} \frac{dy}{dx} = 5y - e^{y} \] Factoring out \( \frac{dy}{dx} \): \[ \left(x e^{y} - 5x - 6y^{2}\right) \frac{dy}{dx} = 5y - e^{y} \] Finally, we can express \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{5y - e^{y}}{x e^{y} - 5x - 6y^{2}} \]