Question

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

• A. $\frac{1}{a(2y+b)}$
• B. $\frac{b}{a(2y+b)}$
• C. $\frac{1}{ab(2y+b)}$
• D. $\frac{ab}{(2y+b)}$

Answer 1

The correct option is B $\frac{b}{a(2y+b)}$

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

$\Rightarrow y=\frac{x}{a+\frac{x}{b+\left[\frac{x}{a+\frac{x}{b+...+\infty }}\right]}}$

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

$\Rightarrow ay+\frac{xy}{b+y}=x$

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

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

Differentiating both sides we get

$ab\cdot dy+2ay\cdot dy=b\cdot dx$

$\Rightarrow (ab+2ay)dy=b\cdot dx$

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

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