Given that, #y=sin^-1{5/13*x+12/13*sqrt(1-x^2)}#.
Observe that, in order that #y# be meaningful, we must have,
#(1-x^2)ge0, or, |x| le1#.
Hence, if we subst. #x=sintheta#, it is a valid substn.
Also, note that, #5/13=cos alpha rArr sin alpha=12/13#.
Thus, with these #x, cos alpha and sin alpha#, we have,
#5/13*x+12/13*sqrt(1-x^2)#,
#=cosalpha*sintheta+sinalphasqrt(1-sin^2theta)#,
#=sinthetacosalpha+costhetasinalpha#,
#=sin(theta+alpha)#.
#:. y=sin^-1{5/13*x+12/13*sqrt(1-x^2)}#,
#=sin^-1{sin(theta+alpha)}#,
#=theta+alpha, (alpha"=constant)"#,
#:. y=sin^-1x+alpha, (alpha"=constant)"#,
#rArr dy/dx=d/dx{sin^-1x+alpha}, (alpha"=constant)"#,
#=1/sqrt(1-x^2)+0#.
#:. dy/dx=1/sqrt(1-x^2)#.
