Question

sin^-1(x (x)^1/2), 0≤x≤1

Answer 1

question is -----> differentiate with respect to x
$\bf{sin^{-1}(x\sqrt{x}),0\leq x\leq1}$

Let $\bf{y=sin^{-1}(x\sqrt{x})}$
differentiate with respect to x,
$\bf{\frac{dy}{dx}=\frac{1}{\sqrt{1-(x\sqrt{x})^2}}\frac{d\{x\sqrt{x}\}}{dx}}\\\\=\bf{\frac{1}{\sqrt{1-x^3}}\frac{3}{2}x^{3/2-1}}\\\\=\bf{\frac{3}{2}\frac{x^{1/2}}{\sqrt{1-x^3}}}\\\\=\bf{\frac{3}{2}\sqrt{\frac{x}{1-x^3}}}$