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Question: The first derivative of the function \(\left\lbrack \cos^{- 1}\left( \sin\frac{\sqrt{1 + x}}{2} \rig...

The first derivative of the function [cos⁡−1(sin⁡1+x2)+xx]\left\lbrack \cos^{- 1}\left( \sin\frac{\sqrt{1 + x}}{2} \right) + x^{x} \right\rbrack with respect to x at x =1 is

A

34\frac{3}{4}

B

0

C

12\frac{1}{2}

D

−12- \frac{1}{2}

Answer

34\frac{3}{4}

Explanation

Solution

f(x)=cos⁡−1[cos⁡(π2−1+x2)]+xx=π2−1+x2+xxf(x) = \cos^{- 1}\left\lbrack \cos\left( \frac{\pi}{2} - \sqrt{\frac{1 + x}{2}} \right) \right\rbrack + x^{x} = \frac{\pi}{2} - \sqrt{\frac{1 + x}{2}} + x^{x}

∴ f′(x)=−12.121+x+xx(1+log⁡x)f^{'}(x) = - \frac{1}{\sqrt{2}}.\frac{1}{2\sqrt{1 + x}} + x^{x}(1 + \log x) ⇒ f′(1)=−14+1=34f^{'}(1) = - \frac{1}{4} + 1 = \frac{3}{4}