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

Find the first order derivative with respect to \( x \) for \( x*e^y=5*x*y+2*y^3 \). Find the first derivative by following steps: - step0: Find the derivative with respect to \(x\): \(xe^{y}=5xy+2y^{3}\) - step1: Take the derivative: \(\frac{d}{dx}\left(xe^{y}\right)=\frac{d}{dx}\left(5xy+2y^{3}\right)\) - step2: Calculate the derivative: \(e^{y}+x\frac{dy}{dx}\times e^{y}=\frac{d}{dx}\left(5xy+2y^{3}\right)\) - step3: Calculate the derivative: \(e^{y}+x\frac{dy}{dx}\times e^{y}=5y+5x\frac{dy}{dx}+6y^{2}\frac{dy}{dx}\) - step4: Rewrite the expression: \(e^{y}+e^{y}x\frac{dy}{dx}=5y+5x\frac{dy}{dx}+6y^{2}\frac{dy}{dx}\) - step5: Collect like terms: \(e^{y}+e^{y}x\frac{dy}{dx}=5y+\left(5x+6y^{2}\right)\frac{dy}{dx}\) - step6: Move the expression to the left side: \(e^{y}x\frac{dy}{dx}-\left(5x+6y^{2}\right)\frac{dy}{dx}=5y-e^{y}\) - step7: Add and subtract: \(\left(e^{y}x-5x-6y^{2}\right)\frac{dy}{dx}=5y-e^{y}\) - step8: Divide both sides: \(\frac{\left(e^{y}x-5x-6y^{2}\right)\frac{dy}{dx}}{e^{y}x-5x-6y^{2}}=\frac{5y-e^{y}}{e^{y}x-5x-6y^{2}}\) - step9: Divide the numbers: \(\frac{dy}{dx}=\frac{5y-e^{y}}{e^{y}x-5x-6y^{2}}\) - step10: Simplify: \(\frac{dy}{dx}=\frac{5y-e^{y}}{xe^{y}-5x-6y^{2}}\) The derivative of \( y \) with respect to \( x \) is given by: \[ \frac{dy}{dx} = \frac{5y-e^{y}}{xe^{y}-5x-6y^{2}} \]