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Differentiation — Maths STD 12 Science — Question

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

Here,
$\text{y}=\sqrt{\cos\text{x}+\sqrt{\cos\text{x}+\sqrt{\cos\text{x}+\ .... \text{to }\infty}}}$
$\text{y}=\sqrt{\cos\text{x}+\text{y}}$
Squaring both the sides,
$\text{y}^2=\cos\text{x}+\text{y}$
Differentiating it with respect to x,
$2\text{y}\frac{\text{dy}}{\text{dx}}=-\sin\text{x}+\frac{\text{dy}}{\text{dx}}$
$\frac{\text{dy}}{\text{dx}}(2\text{y}-1)=-\sin\text{x}$
$\frac{\text{dy}}{\text{dx}}=\frac{-\sin\text{x}}{(2\text{y}-1)}$
$\frac{\text{dy}}{\text{dx}}=\frac{\sin\text{x}}{1-2\text{y}}$

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