Answer
$$ y'
=\frac{ \sinh }{2\sqrt{\cosh x+1}}
.$$
Work Step by Step
Since $ y=\sqrt{\cosh x +1}$, then the derivative $ y'$, using the chain rule, is given by
$$ y'=\frac{1}{2} (\cosh x +1)^{-1/2}(\cosh x +1)'=\frac{1}{2} (\cosh x +1)^{-1/2} \sinh x\\
=\frac{ \sinh }{2\sqrt{\cosh x+1}}
.$$
