Question Watch Video If $-2 x^{3}-y=y^{2}$ then find $\frac{d^{2} y}{d x^{2}}$ at the point $(-1,-2)$ in simplest form. Answer Attempt 1 out of 2

Step 1 ：Given the equation \(-2 x^{3}-y=y^{2}\)

Step 2 ：Rearrange the equation to isolate \(y\): \(y^{2}+y=-2 x^{3}\)

Step 3 ：Differentiate both sides of the equation with respect to \(x\): \(2y\frac{dy}{dx}+\frac{dy}{dx}=-6x^{2}\)

Step 4 ：Differentiate both sides of the equation again with respect to \(x\): \(2\frac{dy}{dx}\frac{dy}{dx}+2y\frac{d^{2}y}{dx^{2}}+\frac{d^{2}y}{dx^{2}}=-12x\)

Step 5 ：Simplify the equation: \((\frac{dy}{dx})^{2}+(2y+1)\frac{d^{2}y}{dx^{2}}=-12x\)

Step 6 ：Find the value of \(\frac{dy}{dx}\) at the point \((-1,-2)\): \(\frac{dy}{dx}=-6\)

Step 7 ：Find the value of \(y\) at the point \((-1,-2)\): \(y=-2\)

Step 8 ：Substitute the values of \(\frac{dy}{dx}=-6\), \(y=-2\), and \(x=-1\) into the equation \((\frac{dy}{dx})^{2}+(2y+1)\frac{d^{2}y}{dx^{2}}=-12x\)

Step 9 ：Solve for \(\frac{d^{2}y}{dx^{2}}\): \(36+(-3)\frac{d^{2}y}{dx^{2}}=12\)

Step 10 ：Divide both sides by \(-3\): \(\frac{d^{2}y}{dx^{2}}=8\)