The derivative of $\left[\frac{{\sin }^{-1}x}{1+{\sin }^{-1}x}\right]$ with respective to ${\sin }^{-1}x$ is

            $=\frac{\frac{\left(1+{\sin }^{-1}x\right)\frac{1}{\sqrt{1-x^2}}-{\sin }^{-1}x\frac{1}{\sqrt{1-x^2}}}{{\left(1+{\sin }^{-1}x\right)}^2}}{\frac{1}{\sqrt{1-x^2}}}$

            $=\frac{1+{\sin }^{-1}x-{\sin }^{-1}x}{{\left(1+{\sin }^{-1}x\right)}^2}$

            $=\frac{1}{{\left(1+{\sin }^{-1}x\right)}^2}$