Question

Find the second derivative of (x^2)(y^2)-12x=8 (implicit differentiation).

Answer 1

$\bf x^2y^2-12x=8\\\\ -----------------------------\\\\ 2xy^2+x^22y\cfrac{dy}{dx}-12x=0\implies 2x\left[ y^2+xy\cfrac{dy}{dx}-6 \right]=0 \\\\\\ xy\cfrac{dy}{dx}=6-y^2\implies \boxed{\cfrac{dy}{dx}=\cfrac{6-y^2}{xy}}$

$\bf \textit{now, using the quotient rule to get }\cfrac{dy^2}{dx^2} \\\\\\ \cfrac{dy^2}{dx^2}=\cfrac{-2y\frac{dy}{dx}xy-(6-y^2)\left( y+x\frac{dy}{dx} \right)}{(xy)^2}$

$\bf now\implies \begin{cases} -2y\frac{dy}{dx}xy\\\\ -2y\frac{6-y^2}{xy}xy\\\\ -2y(6-y^2)\\\\ 2y^3-12y \end{cases}\quad \begin{cases} y+x\frac{dy}{dx}\\\\ y+x\frac{6-y^2}{xy}\\\\ y+\frac{6-y^2}{y}\\\\ \frac{y^2+6-y^2}{y}\\\\ \frac{6}{y} \end{cases}$

$\bf \\\\\\ \cfrac{dy^2}{dx^2}=\cfrac{2y^3-12y-(6-y^2)\frac{6}{y}}{x^2y^2} \\\\\\ \cfrac{dy^2}{dx^2}=\cfrac{2y^3-12y-\frac{36-6y^2}{y}}{x^2y^2} \\\\\\ \cfrac{dy^2}{dx^2}=\cfrac{2y^4-12y^2-36+6y^2}{y}\cdot \cfrac{1}{x^2y^2} \\\\\\ \cfrac{dy^2}{dx^2}=\cfrac{2y^4-6y^2-36}{x^2y^2}$