The first derivative of the function left[ co | vedclass

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(A) Let $f(x) = \cos^{-1}\left( \sin \sqrt{\frac{1+x}{2}} ight) + x^x$. Using the identity $\sin 	heta = \cos(\frac{\pi}{2} - 	heta)$,we have: $f(

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$3/4$

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(A) Let $f(x) = \cos^{-1}\left( \sin \sqrt{\frac{1+x}{2}} ight) + x^x$. Using the identity $\sin 	heta = \cos(\frac{\pi}{2} - 	heta)$,we have: $f(x) = \cos^{-1}\left[ \cos \left( \frac{\pi}{2} - \sqrt{\frac{1+x}{2}} ight) ight] + x^x$. Simplifying the expression: $f(x) = \frac{\pi}{2} - \sqrt{\frac{1+x}{2}} + x^x$. Differentiating with respect to $x$: $f'(x) = 0 - \frac{1}{\sqrt{2}} \cdot \frac{1}{2\sqrt{1+x}} + \frac{d}{dx}(x^x)$. Since $\frac{d}{dx}(x^x) = x^x(1 + \ln x)$,we get: $f'(x) = -\frac{1}{2\sqrt{2(1+x)}} + x^x(1 + \ln x)$. At $x = 1$: $f'(1) = -\frac{1}{2\sqrt{2(1+1)}} + 1^1(1 + \ln 1) = -\frac{1}{2\sqrt{4}} + 1(1 + 0) = -\frac{1}{4} + 1 = \frac{3}{4}$.

The first derivative of the function $\left[ \cos^{-1}\left( \sin \sqrt{\frac{1+x}{2}} \right) + x^x \right]$ with respect to $x$ at $x = 1$ is

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