First of all, it should be clear that the angles in advanced mathematics are all measured by radians. For example, the interval in your topic should be [π/6, 5π/6].

Back to your question, we only need to prove three points: ① Take any point in the interval (pay attention to arbitrariness) and prove that the function is differentiable at that point; ② If it is a left closed interval, it is proved that the function has a right derivative at the left end of the interval; ③ If it is a right closed interval, it is proved that the function has a left derivative at the right end of the interval.