Differentiate the following function with respect to `x :` `sin^(-1){(sinx+cosx)/(sqrt(2))}`,`-(3pi)/4 lt xlt pi/4 `

Explanation

Let `y = sin^-1{(sinx+cosx)/sqrt2}`

`=> y = sin^-1{(sinx*1/sqrt2+cosx*1/sqrt2)}`

`=> y = sin^-1{(sinxcospi/4+cosxsinpi/4)}`

As, `sinAcosB+cosAsinB = sin(A+B)`

`:. y = sin^-1{(sin(x+pi/4)}`

As, `-(3pi)/4 lt x lt pi/4`,

`:. -pi/2 lt x+pi/4 lt pi/2`

So, `x+pi/4` in `sin^-1` range.

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