y=sin-11+x+1-x2 Put x=cos2θ y=sin-12cos2θ+2sin2θ2 =sin-112cosθ+12sinθ =sin-1sinθ+π4 =θ+π4 y=12 cos-1x+π4 dydx=12·-11-x2

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$\frac{d}{dx} [\sin^{- 1} (\frac{\sqrt{1 + x} + \sqrt{1 - x}}{2})] =$

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$\frac{1}{\sqrt{1 - x^{2}}}$

$\frac{- 1}{\sqrt{1 - x^{2}}}$

$\frac{- 1}{2 \sqrt{1 - x^{2}}}$

$\frac{1}{2 \sqrt{1 - x^{2}}}$

answer is C.

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Detailed Solution

$y = \sin^{- 1} (\frac{\sqrt{1 + x} + \sqrt{1 - x}}{2}) Put x = \cos 2 θ y = \sin^{- 1} (\frac{\sqrt{2 \cos^{2} θ} + \sqrt{2 \sin^{2} θ}}{2}) = \sin^{- 1} (\frac{1}{\sqrt{2}} cosθ + \frac{1}{\sqrt{2}} sinθ) = \sin^{- 1} (\sin (θ + \frac{π}{4})) = θ + \frac{π}{4} y = \frac{1}{2} \cos^{- 1} x + \frac{π}{4} \frac{dy}{dx} = \frac{1}{2} \cdot \frac{- 1}{\sqrt{1 - x^{2}}}$