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  4. Find the first derivative of the function [ cos -1( sin √(1+x/2))+xx] with respect to x at x=1.

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Q. Find 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 $x=1$.

Continuity and Differentiability

Solution:

$y =\frac{\pi}{2}-\sin ^{-1} \sin \sqrt{\frac{1+x}{2}}+x^{x}$
$=\frac{\pi}{2}-\sqrt{\frac{1+x}{2}}+x^{x}$
$\frac{d y}{d x}=0-\frac{1 \times(1 / 2)}{2 \sqrt{\frac{1+x}{2}}}+x^{x}(1+\log x)$
at $x =1$
$=-\frac{1}{4}+1=3 / 4$