Verified answer

1) y=x+ctg(1/x)

у' = 1  +  1/Sin²(1/x)  * 1/x² = 1 + 1/(x²*sin(1/x) )

y'= 1 + x⁻²*Sin⁻¹(1/x)

y'' = (x⁻²)' *Sin⁻¹(1/x)  + x⁻²*(Sin⁻¹(1/x))' =

=  -2x⁻³*Sin(1/x) + x⁻²* (-Sin⁻²(1/x)*Cos(1/x)*(-1/x²) ) =

=-2Sin(1/x) /x³ +Cos(1/x)/ (x²*Sin²(1/x))

2) y= 3/x*arcSinx/3

y' =(3/x)' *arcSinx/3 + 3/x*(arcSinx/3)' =

=  -3/х² * arcSinx/3 + 3/x*1/√(1 - x²/9) * (1/3)=

=-3arcSinx/3 /х² + 1/√(1 - х²/9)

y'' = (-x²/√(1 - x²/9)  -6x*arcSinx/3 )/x⁴