Solution:

Now let’s consider $y=\sin ^{-1}(x \sqrt{x})$

or $y=\frac{\sin ^{-1}\left(x^{\frac{3}{2}}\right)}{}$

Now derivating the above given function:

$\frac{d y}{d x}=\frac{1}{\sqrt{1-\left(x^{\frac{3}{2}}\right)^{2}}} \frac{d}{d x} x^{\frac{3}{2}}$

$=\frac{1}{\sqrt{1-x^{3}}} \cdot \frac{3}{2} x^{\frac{1}{2}}$

$=\frac{3}{2} \sqrt{\frac{x}{1-x^{3}}}$