Question

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

• A. $\frac{a}{ab+2ay}$
• B. $\frac{b}{ab+2by}$
• C. $\frac{a}{ab+2by}$
• D. $\frac{b}{ab+2ay}$

Answer 1

The correct option is D $\frac{b}{ab+2ay}$

$y=\frac{x}{a+\frac{x}{b+y}}$
$\Rightarrow y=\frac{x(b+y)}{ab+ay+x}$
$\Rightarrow aby+a{y}^2+xy=bx+xy\Rightarrow aby+a{y}^2=bx$
Differentiating both sides w.r.t $x$
$\Rightarrow (ab+2ay)\frac{dy}{dx}=b\therefore \frac{dy}{dx}=\frac{b}{ab+2ay}$