Hoạt động 3 trang 65 SGK Toán 11 Tập 2 Cánh diều - CD

Ta có:

\(\begin{array}{l}f'({x_0}) = \mathop {\lim }\limits_{x \to {x_0}} \frac{{f(x) - f({x_0})}}{{x - {x_0}}} = \mathop {\lim }\limits_{x \to {x_0}} \frac{{\sin x - \sin {x_0}}}{{x - {x_0}}} = \mathop {\lim }\limits_{x \to {x_0}} \frac{{2\,.\,\cos x\frac{{x + {x_0}}}{2}.\sin \frac{{x - {x_0}}}{2}}}{{x - {x_0}}}\\ = \mathop {\lim }\limits_{x \to {x_0}} \frac{{2.\frac{{x - {x_0}}}{2}.\cos \frac{{x + {x_0}}}{2}}}{{x - {x_0}}} = \mathop {\lim }\limits_{x \to {x_0}} \,\cos \frac{{x + {x_0}}}{2} = \cos \frac{{2{x_0}}}{2} = \cos {x_0}\end{array}\)

Suy ra \((\sin x)' = \cos x\).

-- Mod Toán 11 HỌC247