Производная равна:

y ' = (ln sin^2 2x) ' = 1/(sin^2 2x) * (sin^2 2x) ' = 1/(sin^2 2x) * 2 * sin 2x * (sin 2 * x) ' = 2/sin 2x * cos 2x * 2 = 4/sin 2x * cos 2x = 4 * ctg 2x.