Question

Differentiate w.r.t. \(x\) the function: \(sin^{-1}(x\sqrt{x}),0≤x≤1\)

Answer 1

The correct answer is \(=\frac{3}{2}\sqrt{\frac{x}{1-x^3}}\)
Let \(y=sin^{-1}(x\sqrt{x})\)
Using chain rule,we obtain
\(\frac{dy}{dx}=\frac{d}{dx}(sin^{-1}(x\sqrt{x}))\)
\(=\frac{1}{\sqrt{1-(x\sqrt{x})^2}}\times\frac{d}{dx}(x\sqrt{x})\)
\(=\frac{1}{\sqrt{1-x^3}}.\frac{d}{dx}(x^{\frac{3}{2}})\)
\(=\frac{1}{\sqrt{1-x^3}}\times\frac{3}{2}.x^{\frac{1}{2}}\)
\(=\frac{3\sqrt{x}}{2\sqrt{1-x^3}}\)
\(=\frac{3}{2}\sqrt{\frac{x}{1-x^3}}\)