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Question

If $$ \displaystyle y=\dfrac { x }{ a+\dfrac { x }{ b+y }  } $$, then $$ \displaystyle \frac{dy}{dx}$$ is



A
aab+2ay
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B
bab+2by
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C
aab+2by
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D
bab+2ay
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Solution

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

$$\displaystyle y=\dfrac { x }{ a+\dfrac { x }{ b+y }  }$$

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

$$ \displaystyle \Rightarrow $$ $$aby+ay^{2}+xy = xb+xy$$

$$ \displaystyle \Rightarrow $$ $$aby+ay^{2} = xb$$

Differentiating both side w.r.t $$x$$ we get,

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


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