If $y=\frac{x}{a+\frac{x}{b+\frac{x}{a+\frac{x}{b+\frac{x}{a+...∞}}}}}$, then $\frac{dy}{dx}$ is:

$\frac{b}{ab+2ay}$

We have, $y=\frac{x}{a+\frac{x}{b+\frac{x}{a+\frac{x}{b+\frac{x}{a+...∞}}}}}$

$⇒y=\frac{x}{a+\frac{x}{b+y}}⇒y=\frac{x(b+y)}{ab+ay+x}$

$⇒aby+ay^2+xy=xb+xy⇒aby+ay^2=xb$

On differentiating w.r.t. x, we get :

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

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