use implicit differentiation to find $\frac{d^{2}y}{dx^{2}}$.\n$4x^{2}=2y^{3}+4y - 2$\na. $\frac{4(3y^{2}+2)^…

Explanation:

Step1: Differentiate both sides with respect to x

Differentiate $4x^{2}=2y^{3}+4y - 2$ using the chain - rule. The derivative of $4x^{2}$ with respect to $x$ is $8x$. The derivative of $2y^{3}$ with respect to $x$ is $2\times3y^{2}\frac{dy}{dx}=6y^{2}\frac{dy}{dx}$, the derivative of $4y$ with respect to $x$ is $4\frac{dy}{dx}$, and the derivative of $-2$ with respect to $x$ is $0$. So, $8x = 6y^{2}\frac{dy}{dx}+4\frac{dy}{dx}$. Factor out $\frac{dy}{dx}$: $8x=(6y^{2} + 4)\frac{dy}{dx}$, then $\frac{dy}{dx}=\frac{8x}{6y^{2}+4}=\frac{4x}{3y^{2}+2}$.

Step2: Differentiate $\frac{dy}{dx}$ with respect to x to find $\frac{d^{2}y}{dx^{2}}$

Use the quotient - rule $\left(\frac{u}{v}\right)'=\frac{u'v - uv'}{v^{2}}$, where $u = 4x$ and $v=3y^{2}+2$. $u' = 4$ and $v'=6y\frac{dy}{dx}$. Substitute $\frac{dy}{dx}=\frac{4x}{3y^{2}+2}$ into $v'$: $v'=6y\times\frac{4x}{3y^{2}+2}=\frac{24xy}{3y^{2}+2}$. Then $\frac{d^{2}y}{dx^{2}}=\frac{4(3y^{2}+2)-4x\times\frac{24xy}{3y^{2}+2}}{(3y^{2}+2)^{2}}$. Simplify the numerator: [ \begin{align*} 4(3y^{2}+2)-\frac{96x^{2}y}{3y^{2}+2}&=\frac{4(3y^{2}+2)^{2}-96x^{2}y}{3y^{2}+2}\ \end{align*} ] So, $\frac{d^{2}y}{dx^{2}}=\frac{4(3y^{2}+2)^{2}-96x^{2}y}{(3y^{2}+2)^{3}}$.

Answer:

A. $\frac{4(3y^{2}+2)^{2}-96x^{2}y}{(3y^{2}+2)^{3}}$