[tex]\bf x^2y^2-12x=8\\\\
-----------------------------\\\\
2xy^2+x^22y\cfrac{dy}{dx}-12x=0\implies 2x\left[ y^2+xy\cfrac{dy}{dx}-6 \right]=0
\\\\\\
xy\cfrac{dy}{dx}=6-y^2\implies \boxed{\cfrac{dy}{dx}=\cfrac{6-y^2}{xy}}[/tex]
[tex]\bf \textit{now, using the quotient rule to get }\cfrac{dy^2}{dx^2}
\\\\\\
\cfrac{dy^2}{dx^2}=\cfrac{-2y\frac{dy}{dx}xy-(6-y^2)\left( y+x\frac{dy}{dx} \right)}{(xy)^2}
\\\\\\[/tex]
[tex]\bf now\implies
\begin{cases}
-2y\frac{dy}{dx}xy\\\\
-2y\frac{6-y^2}{xy}xy\\\\
-2y(6-y^2)\\\\
2y^3-12y
\end{cases}\quad
\begin{cases}
y+x\frac{dy}{dx}\\\\
y+x\frac{6-y^2}{xy}\\\\
y+\frac{6-y^2}{y}\\\\
\frac{y^2+6-y^2}{y}\\\\
\frac{6}{y}
\end{cases}[/tex]
[tex]\bf \\\\\\
\cfrac{dy^2}{dx^2}=\cfrac{2y^3-12y-(6-y^2)\frac{6}{y}}{x^2y^2}
\\\\\\
\cfrac{dy^2}{dx^2}=\cfrac{2y^3-12y-\frac{36-6y^2}{y}}{x^2y^2}
\\\\\\
\cfrac{dy^2}{dx^2}=\cfrac{2y^4-12y^2-36+6y^2}{y}\cdot \cfrac{1}{x^2y^2}
\\\\\\
\cfrac{dy^2}{dx^2}=\cfrac{2y^4-6y^2-36}{x^2y^2}[/tex]
