Differentiate the following function with respect to `x :` `sin^(-1){(x+sqrt(1-x^2))/(sqrt(2))},-1

Explanation

Solution:

Let `y=\sin ^{-1}\{\frac{x+\sqrt{1-x^{2}}}{\sqrt{2}}\}`

Let` x=\sin t`

Then,

`y=\sin ^{-1}\{\frac{\sin t+\sqrt{1-\sin ^{2} t}}{\sqrt{2}}\} `

`\Rightarrow y=\sin ^{-1}\{\frac{\sin t+\cos t}{\sqrt{2}} \} `

`\Rightarrow y=\sin ^{-1}\{(\sin t \cdot \frac{1}{\sqrt{2}}+\cos t \cdot \frac{1}{\sqrt{2}})\} `

`\Rightarrow y=\sin ^{-1}\{(\sin t \frac{\cos \pi}{4}+\cos t \frac{\sin \pi}{4}}`

...