Question

implicit differentiation of x^2=(6x+6y)/(6x-6y)

Answer 1

$\textbf{Given:}$

$x^2=\frac{6x+6y}{6x-6y}$

$\implies\;(6x-6y)x^2=6x+6y$

$\text{Differentiate with respect to x}$

$(6x-6y)\;2x+x^2(6-6\frac{dy}{dx})=6+6\frac{dy}{dx}$

$12x^2-12xy+6x^2-6x^2\frac{dy}{dx}=6+6\frac{dy}{dx}$

$18x^2-12xy-6=6\frac{dy}{dx}+6x^2\frac{dy}{dx}$

$18x^2-12xy-6=6\frac{dy}{dx}[1+x^2]$

$3x^2-2xy-1=\frac{dy}{dx}[1+x^2]$

$\frac{3x^2-2xy-1}{1+x^2}=\frac{dy}{dx}$

$\therefore\bf\frac{dy}{dx}=\frac{3x^2-2xy-1}{1+x^2}$