Let F$($x$)$ be defined asF$($x$)=$ integral from $\{$x$\}$ to $\{$arcsin x$\}$ of: cos$^2($t$)$ dt

$($a$)$ Compute the derivative F$\text{'}($x$)$

$($b$)$ Simplify your answer in part $($a$)$ completely. That is$,$ do not leave an inverse trig inside a trig function.