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Differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentiation3 Marks
Question
If $\text{y}=\sqrt{\text{x}+\sqrt{\text{x}+\sqrt{\text{x}+\ .... \text{to }\infty}}},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{1}{2\text{y}-1}$
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Answer

We have, $\text{y}=\sqrt{\text{x}+\sqrt{\text{x}+\sqrt{\text{x}+\ .... \text{to }\infty}}}$
Squaring both sides, we get,
$y^2 = x + y$
$\Rightarrow 2\text{y}\frac{\text{dy}}{\text{dx}}=1+\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}(2\text{y}-1)=1$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{1}{2\text{y}-1}$

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