Differentiate `sin(-1)[(sqrt(1+x)+sqrt(1-x))/2]` with respect to 'x'.

Explanation

Let y=`sin(-1)[(sqrt(1+x)+sqrt(1-x))/2]` with respect to 'x'.
and `x=cos theta`
`therefore y=sin^(-1)[(sqrt(1+ cos theta)+sqrt(1-costheta))/(2)]`
`sin^(-1)[(sqrt(2+ cos^2""theta/2)+sqrt(2 sin ^2 ""theta/2))/(2)]`
`=sin ^-1[1/sqrt(2)cos""theta/2+1/sqrt(2)sin""theta /2]`
`=sin ^(-1)[sin ""pi/4 cos""theta/2+cos""pi/4sin ""theta/2]`
`=sin ^-1sin(pi/4+theta/2)`
`=pi/4+theta/2=pi/4+1/2cos^-1x`
`rArr dy/dx=d/dx(pi/4+1/2cos^-1x)`
`=0+1/2((-1)/(sqrt(1-x^2)))=(-1)/(2sqrt(1-x^2)).`