Answer
$$y'=\frac{1}{ \sqrt{1- x^2}\sin^{-1}x}.$$
Work Step by Step
Since $ y=\ln(arcsin \ x)=\ln(\sin^{-1}x) $, then
$$ y'=\frac{1}{\sin^{-1}x}(\sin^{-1}x)'=\frac{1}{\sin^{-1}x}\frac{1}{ \sqrt{1- x^2}}
\\=\frac{1}{ \sqrt{1- x^2}\sin^{-1}x}.$$