If y = x left[ left( cos frac{x}{2} + sin fra | vedclass

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(A) Given $y = x \left[ \left( \cos \frac{x}{2} + \sin \frac{x}{2} \right) \left( \cos \frac{x}{2} - \sin \frac{x}{2} \right) + \sin x \right] + \frac

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$(1 + x)\cos x + (1 - x)\sin x - \frac{1}{4x\sqrt{x}}$

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(A) Given $y = x \left[ \left( \cos \frac{x}{2} + \sin \frac{x}{2} \right) \left( \cos \frac{x}{2} - \sin \frac{x}{2} \right) + \sin x \right] + \frac{1}{2\sqrt{x}}$.Using the identity $(a+b)(a-b) = a^2 - b^2$,we have $\left( \cos \frac{x}{2} + \sin \frac{x}{2} \right) \left( \cos \frac{x}{2} - \sin \frac{x}{2} \right) = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} = \cos x$.Substituting this into the expression for $y$:$y = x(\cos x + \sin x) + \frac{1}{2}x^{-1/2}$.Differentiating with respect to $x$ using the product rule:$\frac{dy}{dx} = \frac{d}{dx}[x(\cos x + \sin x)] + \frac{d}{dx}(\frac{1}{2}x^{-1/2})$.$\frac{dy}{dx} = x(-\sin x + \cos x) + 1(\cos x + \sin x) + \frac{1}{2} \cdot (-\frac{1}{2})x^{-3/2}$.$\frac{dy}{dx} = -x\sin x + x\cos x + \cos x + \sin x - \frac{1}{4x\sqrt{x}}$.$\frac{dy}{dx} = (1 + x)\cos x + (1 - x)\sin x - \frac{1}{4x\sqrt{x}}$.

If $y = x \left[ \left( \cos \frac{x}{2} + \sin \frac{x}{2} \right) \left( \cos \frac{x}{2} - \sin \frac{x}{2} \right) + \sin x \right] + \frac{1}{2\sqrt{x}}$,then $\frac{dy}{dx} = $

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