Question

Differentiate $x=y+\frac{1}{y+\frac{1}{y+\frac{1}{y+\frac{1}{y+\frac{1}{y+...\infty }}}}}$ w.r.t $x$

Answer 1

$x=y+\frac{1}{y+\frac{1}{y+\frac{1}{y+\frac{1}{y}.......\infty }}}$

$\Rightarrow y+\frac{1}{x}=x$

$\Rightarrow xy=x^2-1$

$\Rightarrow x\frac{dy}{dx}+y=2x$

$\Rightarrow x\frac{dy}{dx}=2x-y$

$\Rightarrow \frac{dy}{dx}=\frac{2x-y}{x}$