Differentiate the function sin ^ -1 ( x x ) , 0 x 1 w.r.t. to x. - Vidyadip

Question

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

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Answer

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

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

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

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

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

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