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  4. If y = sin-1 √ 1- x + cos-1 √x, then (dy/dx) equals

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Q. If $y = \sin^{-1} \sqrt{ 1- x} + \cos^{-1} \sqrt{x}$, then $\frac{dy}{dx}$ equals

Limits and Derivatives

Solution:

$y =\sin^{-1} \sqrt{1-x} + \cos^{-1}\sqrt{x} $
$\therefore \frac{dy}{dx} = \frac{1}{\sqrt{1-\left(1-x\right)} }. \frac{d}{dx} \sqrt{1-x} - \frac{1}{\sqrt{1-x}} \frac{d}{dx} \left(\sqrt{x}\right) $
$= \frac{1}{\sqrt{x}} . \frac{1\left(-1\right)}{2\sqrt{1-x}} - \frac{1}{\sqrt{1-x}} . \frac{1}{2\sqrt{x}}$
$ = - \frac{1}{2\sqrt{x}\sqrt{1-x}} - \frac{1}{2\sqrt{x}\sqrt{1-x}} $
$=- \frac{1}{\sqrt{x} \sqrt{1-x}} = - \frac{1}{\sqrt{x\left(1-x\right)}}$