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

Option: 1

3/4


Option: 2

0


Option: 3

1/2


Option: 4

-1/2


Answers (1)

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\begin{aligned} & f(x)=\cos ^{-1}\left[\cos \left(\frac{\pi}{2}-\sqrt{\frac{1+x}{2}}\right)\right]+x^x=\frac{\pi}{2}-\sqrt{\frac{1+x}{2}}+x^x \\ & \therefore \quad f^{\prime}(x)=-\frac{1}{\sqrt{2}} \cdot \frac{1}{2 \sqrt{1+x}}+x^x(1+\log x) \Rightarrow f^{\prime}(1)=-\frac{1}{4}+1=\frac{3}{4} \end{aligned}

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