Вторая производная от y=ln (tgx)

y' = (ln (tgx)) = (1/tgx) * (tgx) ') = = (1/tgx) * (1/sin²x) = 1 / (sinx*cos) =

=2*1 / (2*sinx*cosx) = 2/sin (2x) = 2 / (sin (2x)).

y'' = (2 * / (sin (2x)) ' = ((2) '*sin (2x) - 2 * (-sin (2x)) ') / sin² (2x) = - 2*cos (2x) * 2/sin² (2x) =

=-4 * (cos²x-sin²x) / (2*sinx*cosx) ²=-4 * (cos²x-sin²x) / (4*sin²x*cos²x) =

= - ((1/sin²x) - (1/cos²x)) = - (csc² (x) - sec² (x)) = sec² (x) - csc² (x),