Question

find the deriavative of

${\sin}^{ - 1} (x) + \: { \sin}^{ - 1} ( \sqrt{1 - {x}^{2} } )$

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Answer 1

Step-by-step explanation:

We have,

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

$Let\: x = \sin(\alpha)$

$\implies \: y = \sin^{ - 1} ( \sin( \alpha ) ) + \sin^{ - 1} ( \sqrt{1 - \sin^{2} ( \alpha ) } )$

$\implies \: y = \alpha + \sin ^{ - 1} ( \cos( \alpha ) )$

$\implies \: y = \alpha + \frac{\pi}{2} - \sin ^{ - 1} ( \sin( \alpha ) )$

$\implies \: y = \alpha + \frac{\pi}{2} - \alpha$

$\implies \: y = \frac{\pi}{2}$

$\implies \frac{dy}{dx} = 0$