Differentiate the following functions with respect to $x$ :
$\sin ^{-1}\left\{\frac{x+\sqrt{1-x^{2}}}{\sqrt{2}}\right\},-1
$y=\sin ^{-1}\left\{\frac{x+\sqrt{1-x^{2}}}{\sqrt{2}}\right\}$
Let $x=\sin \theta$
Now
$y=\sin ^{-1}\left\{\frac{\sin \theta+\sqrt{1-\sin ^{2} \theta}}{\sqrt{2}}\right\}$
Using $\sin ^{2} \theta+\cos ^{2} \theta=1$
$y=\sin ^{-1}\left\{\frac{\sin \theta+\cos \theta}{\sqrt{2}}\right\}$
Now
$y=\sin ^{-1}\left\{\sin \theta \frac{1}{\sqrt{2}}+\cos \theta \frac{1}{\sqrt{2}}\right\}$
$y=\sin ^{-1}\left\{\sin \theta \cos \left(\frac{\pi}{4}\right)+\cos \theta \sin \left(\frac{\pi}{4}\right)\right\}$
Using $\sin (A+B)=\sin A \cos B+\cos A \sin B$
$y=\sin ^{-1}\left\{\sin \left(\theta+\frac{\pi}{4}\right)\right\}$
Considering the limits,
$-1 $-1<\sin \theta<1$ $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$ $-\frac{\pi}{2}+\frac{\pi}{4}<\theta+\frac{\pi}{4}<\frac{\pi}{2}+\frac{\pi}{4}$ $-\frac{\pi}{4}<\theta+\frac{\pi}{4}<\frac{3 \pi}{4}$ Now, $y=\sin ^{-1}\left\{\sin \left(\theta+\frac{\pi}{4}\right)\right\}$ $y=\theta+\frac{\pi}{4}$ $y=\sin ^{-1} x+\frac{\pi}{4}$ Differentiating w.r.t $\mathrm{x}$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\sin ^{-1} \mathrm{x}+\frac{\pi}{4}\right)$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\sqrt{1-\mathrm{x}^{2}}}$