Question

If $y=\frac{x}{a+\frac{x}{a+\frac{x}{a+..........\infty }}}$, then find $\frac{dy}{dx}$

Answer 1

Let,

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

Hence,

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

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

$\Rightarrow ay+y^2=x$

$\Rightarrow a\frac{dy}{dx}+2y\frac{dy}{dx}=1$

$\Rightarrow \frac{dy}{dx}(a+2y)=1$

$\Rightarrow \frac{dy}{dx}=\frac{1}{a+2y}=\frac{1}{a+2\times \frac{x}{a+\frac{x}{a+\frac{x}{a+..........\infty }}}}$