If  y=cosx+cosx+⋯+∞, then  dydx=

# If  $y=\sqrt{\mathrm{cos}x+\sqrt{\mathrm{cos}x+\cdots +\infty }}$, then  $\frac{dy}{dx}=$

1. A

$\frac{\mathrm{sin}x}{2y-1}$

2. B

$\frac{\mathrm{cos}x}{2y-1}$

3. C

$\frac{\mathrm{cos}x}{1-2y}$

4. D

$\frac{\mathrm{sin}x}{1-2y}$

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### Solution:

$y=\sqrt{\mathrm{cos}x+y}$

Squaring on both sides

${y}^{2}=\mathrm{cos}x+y$

$2y.\frac{dy}{dx}=-\mathrm{sin}x+\frac{dy}{dx}$

$\left(2y-1\right)\frac{dy}{dx}=-\mathrm{sin}x$

$\frac{dy}{dx}=\frac{-\mathrm{sin}x}{2y-1}$

$\frac{dy}{dx}=\frac{\mathrm{sin}x}{1-2y}$

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