Differentiate the following functions with respect to x :

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 $\mathrm{x}$,

$y=x+\frac{\pi}{4}$

$\frac{\mathrm{dy}}{\mathrm{dx}}=1$

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