Question

Найти значение производной функции в точке :
y=ctg(pi/6-x) если x^0=pi/3

Answer 1

$y = \text{ctg} \left(\dfrac{\pi}{6} - x \right)$

Найдем производную:

$y' = \left(\text{ctg} \left(\dfrac{\pi}{6} - x \right) \right)' = -\dfrac{1}{\sin^{2}\left(\dfrac{\pi}{6} - x \right)} \cdot \left(\dfrac{\pi}{6} - x \right)' = \dfrac{1}{\sin^{2}\left(\dfrac{\pi}{6} - x \right)}$

Найдем значение производной в точке $x_{0} = \dfrac{\pi}{3}$

$y'_{0} = \dfrac{1}{\sin^{2}\left(\dfrac{\pi}{6} - \dfrac{\pi}{3} \right)} = \dfrac{1}{\sin^{2}\left(- \dfrac{\pi}{6} \right)} = \dfrac{1}{\sin^{2}\dfrac{\pi}{6}} =\dfrac{1}{\left( \dfrac{1}{2} \right)^{2}} = 4$

Ответ: 4.