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Algebra of Derivatives

If y= x / a...

Question

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

A

\frac{a}{a b + 2 a y}

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B

\frac{b}{a b + 2 b y}

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C

\frac{a}{a b + 2 b y}

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D

\frac{b}{a b + 2 a y}

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Solution

The correct option is D $\frac{b}{a b + 2 a y}$
$y = \frac{x}{a + \frac{x}{b + y}}$
$\Rightarrow y = \frac{x (b + y)}{a b + a y + x}$
$\Rightarrow a b y + a y^{2} + x y = b x + x y \Rightarrow a b y + a y^{2} = b x$
Differentiating both sides w.r.t $x$
$\Rightarrow (a b + 2 a y) \frac{d y}{d x} = b ∴ \frac{d y}{d x} = \frac{b}{a b + 2 a y}$

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