Question

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 $x=1$.

Answer 1

Answer: 0.75

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