If $y=\frac{x}{a+\frac{x}{b+\frac{x}{a+\frac{x}{b+\dots \dots \dots ..+\text{ to }\mathrm{\infty }}}}}\text{ then }\frac{dy}{dx}=$

$\text{ Here, }y=\frac{x}{a+\frac{x}{b+y}}=\frac{x(b+y)}{ab+ay+x}$

$\Rightarrow aby+a{y}^2+xy=bx+xy$

$\Rightarrow aby+a{y}^2=bx$

$\text{ On differentiating w.r.t. }x\text{, we get }$

$ab\frac{dy}{dx}+2ay\frac{dy}{dx}=b$

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