Question

Differentiate the following w.r.t x if x € (0,1).
$\sf y = {sin}^{ - 1} (x \sqrt{1 - x} - \sqrt{x} \sqrt{1 - {x}^{2} } )$
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Answer 1

Answer in the attachment....

Hope it helps you ✌️

Answer 2

✪GIVEN:- $\sf y = {sin}^{ - 1} (x \sqrt{1 - x} - \sqrt{x} \sqrt{1 - {x}^{2} } )$ & x € (0,1).

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✪TO FIND :- Differentiation of the following wrt x.

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✍SOLUTION :-

$= > y = \sin {}^{ - 1} (x \sqrt{1 - x} \: - \sqrt{x} \sqrt{1 - x {}^{2} } )$

$= > y = \sin {}^{ - 1} (x \sqrt{1 - ( \sqrt{x} ) {}^{2} } - \sqrt{x} \sqrt{1 - x {}^{2} } )$

$✪ \boxed{ \sin {}^{ - 1} ( a ) - \sin {}^{ - 1} (b) = \sin {}^{ - 1} (a \sqrt{1 - b {}^{2} } - b \sqrt{1 - a {}^{2} } ) }$

$= > \: y = \sin {}^{ - 1} (x) - \sin {}^{ - 1} ( \sqrt{x} )$

$= > \: \frac{dy}{dx} = \frac{1}{ \sqrt{1 - x {}^{2} } } - \frac{1}{ \sqrt{1 - ( \sqrt{x} ) {}^{2} } }$

$= > \frac{1}{ \sqrt{1 - x {}^{2} } } - \frac{1}{ \sqrt{1 - x} } ( \frac{1}{2 \sqrt{x} } )$

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

Hence, the Answer is,

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

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HOPE IT HELPS :)