Find \( y^{\prime \prime} \) by implicit differentiation. Simplify where possible. \( x^{2}+3 y^{2}=3 \) Need Help? Read It Watch It SIRMIT ANSWED

Find the second order derivative with respect to \( x \) for \( x^{2}+3y^{2}=3 \). Find the second derivative by following steps: - step0: Find the second derivative with respect to \(x\): \(x^{2}+3y^{2}=3\) - step1: Rewrite the expression: \(\frac{dy}{dx}=-\frac{x}{3y}\) - step2: Take the derivative: \(\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(-\frac{x}{3y}\right)\) - step3: Calculate the derivative: \(\frac{d^2y}{dx^2}=\frac{d}{dx}\left(-\frac{x}{3y}\right)\) - step4: Use differentiation rules: \(\frac{d^2y}{dx^2}=-\frac{\frac{d}{dx}\left(x\right)\times 3y-x\times \frac{d}{dx}\left(3y\right)}{\left(3y\right)^{2}}\) - step5: Calculate the derivative: \(\frac{d^2y}{dx^2}=-\frac{1\times 3y-x\times \frac{d}{dx}\left(3y\right)}{\left(3y\right)^{2}}\) - step6: Calculate the derivative: \(\frac{d^2y}{dx^2}=-\frac{1\times 3y-x\times 3\frac{dy}{dx}}{\left(3y\right)^{2}}\) - step7: Calculate: \(\frac{d^2y}{dx^2}=-\frac{3y-x\times 3\frac{dy}{dx}}{\left(3y\right)^{2}}\) - step8: Calculate: \(\frac{d^2y}{dx^2}=-\frac{3y-3x\frac{dy}{dx}}{\left(3y\right)^{2}}\) - step9: Calculate: \(\frac{d^2y}{dx^2}=-\frac{3y-3x\frac{dy}{dx}}{9y^{2}}\) - step10: Calculate: \(\frac{d^2y}{dx^2}=-\frac{y-x\frac{dy}{dx}}{3y^{2}}\) - step11: Substitute back: \(\frac{d^2y}{dx^2}=-\frac{y-x\left(-\frac{x}{3y}\right)}{3y^{2}}\) - step12: Calculate: \(\frac{d^2y}{dx^2}=-\frac{3y^{2}+x^{2}}{9y^{3}}\) The second derivative of \( y \) with respect to \( x \) is \( \frac{d^2y}{dx^2} = -\frac{3y^{2}+x^{2}}{9y^{3}} \).