Differentiate the following functions with respect to x :

Question:

Differentiate the following functions with respect to $x$ :

$\sin ^{-1}\left\{\sqrt{1-x^{2}}\right\}, 0

Solution:

$y=\sin ^{-1}\left\{\sqrt{1-x^{2}}\right\}$

let $x=\cos \theta$

Now

$y=\sin ^{-1}\left\{\sqrt{1-\cos ^{2} \theta}\right\}$

Using $\sin ^{2} \theta+\cos ^{2} \theta=1$

$y=\sin ^{-1}(\sin \theta)$

Considering the limits,

$0

$0<\cos \theta<1$

$0<\theta<\frac{\pi}{2}$

Now, $y=\sin ^{-1}(\sin \theta)$

$y=\theta$

$y=\cos ^{-1} x$

Differentiating w.r.t $x$, we get

$\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{1}{\sqrt{1-\mathrm{x}^{2}}}$

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