5. \(f(x) = \sin(3x)\), \(x_0 = \frac{\pi}{12}\).

1. Находим производную функции (f(x) = \(\sin\)(3x)):
(f'(x) = 3\(\cos\)(3x))

2. Вычисляем значение производной в точке \(x_0 = \frac{\pi}{12}\):
(f'\(\frac{\pi}{12}\) = 3\(\cos\)\(3 \cdot \frac{\pi}{12}\) = 3\(\cos\)\(\frac{\pi}{4}\) = 3 \(\cdot\) \(\frac{\sqrt{2}}{2}\) = \(\frac{3\sqrt{2}}{2}\))

Ответ: (f'\(\frac{\pi}{12}\) = \(\frac{3\sqrt{2}}{2}\))