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