f(x) = 2ln(sqrt(sin(2x)))
f'(x) = 2 (1 / sqrt(sin(2x))) (cos(2x) 2)
f'(П/8) = 2 (1 / sqrt(sin(2 П/8))) (cos(2 П/8) 2)
f'(П/8) = 2 (1 / sqrt(sin(П/4))) (-1 2)
f'(П/8) = 2 (1 / 1) * (-2)
f'(П/8) = -4

f(x) = arctg(e^-x)
f'(x) = -1 / (1 + (e^-x)^2) (-e^-x)
f'(0) = -1 / (1 + (e^0)^2) (-e^0)
f'(0) = -1 / (1 + 1) * (-1)
f'(0) = -1 / 2

f(x) = sin(2x)(1 + cos(2x))
f'(x) = 2 cos(2x) (1 + cos(2x)) + sin(2x) (-2 sin(2x))
f'(П/4) = 2 cos(2 П/4) (1 + cos(2 П/4)) + sin(2 П/4) (-2 sin(2 П/4))
f'(П/4) = 2 cos(П/2) (1 + cos(П/2)) + sin(П/2) (-2 sin(П/2))
f'(П/4) = 2 0 (1 - 1) + 1 (-2 1)
f'(П/4) = 0 - 2
f'(П/4) = -2

f(x) = ln(tg^2(2x))
f'(x) = 2 (1 / tg^2(2x)) tg(2x) 2
f'(П/8) = 2 (1 / tg^2(2 П/8)) tg(2 П/8) 2
f'(П/8) = 2 (1 / tg^2(П/4)) tg(П/4) 2
f'(П/8) = 2 (1 / 1) 1 2
f'(П/8) = 4