Find the derivative of cos^{-1} sqrt{frac{1 + | vedclass

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(C) Let $y = \cos^{-1} \sqrt{\frac{1 + x^2}{2}}$.Substitute $x^2 = \cos 2	heta$,which implies $	heta = \frac{1}{2} \cos^{-1}(x^2)$.Then,$\sqrt{\frac

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$-\frac{x}{\sqrt{1 - x^4}}$

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(C) Let $y = \cos^{-1} \sqrt{\frac{1 + x^2}{2}}$.Substitute $x^2 = \cos 2	heta$,which implies $	heta = \frac{1}{2} \cos^{-1}(x^2)$.Then,$\sqrt{\frac{1 + x^2}{2}} = \sqrt{\frac{1 + \cos 2	heta}{2}} = \sqrt{\frac{2\cos^2 	heta}{2}} = \cos 	heta$.So,$y = \cos^{-1}(\cos 	heta) = 	heta = \frac{1}{2} \cos^{-1}(x^2)$.Differentiating with respect to $x$:$\frac{dy}{dx} = \frac{1}{2} \cdot \frac{d}{dx}(\cos^{-1}(x^2)) = \frac{1}{2} \cdot \left( -\frac{1}{\sqrt{1 - (x^2)^2}} ight) \cdot \frac{d}{dx}(x^2)$.$\frac{dy}{dx} = \frac{1}{2} \cdot \left( -\frac{1}{\sqrt{1 - x^4}} ight) \cdot 2x = -\frac{x}{\sqrt{1 - x^4}}$.

Find the derivative of $\cos^{-1} \sqrt{\frac{1 + x^2}{2}}$ with respect to $x$.

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