`y=sin^(-1)((5x+12sqrt(1-x^2))/(13))`

Explanation

Let `y=sin^(-1)((5x+12sqrt(1-x^(2)))/(13))`
Put `x = sin theta`. Then `theta = sin^(-1)x`
`therefore y=sin^(-1)((5sintheta+12sqrt(1-sin^(2)theta))/(13))`
`=sin^(-1)((5sintheta+12costheta)/(13))`
`=sin^(-1)[(sintheta)((5)/(13))+(costheta)((12)/(13))]`
Since `((5)/(13))^(2)+((12)/(13))^(2)=(25)/(169)+(144)/(169)=1` we can write
`(5)/(13)=cos alpha and (12)/(13)=sinalpha`
`therefore y=sin^(-1)(sinthetacosalpha+costhetasinalpha)`
`=sin^(-1)[sin(theta+alpha)]=theta+alpha`
`=sin^(-1)x+alpha`, where `alpha` is a constant
`therefore(dy)/(dx)=(d)/(dx)(sin^(-1)x)+(d)/(dx)(alpha)`
`=(1)/(sqrt(1-x^(2)))+0=(1)/(sqrt(1-x^(2)))`