单变量微积分学习笔记：三角函数求导法则（9）【2，6，7，8】

公式

\((\sin(x))' = \cos(x)\)
\((\cos(x))' = -\sin(x)\)
\((\tan(x))' = \sec^2(x)\)

证明

\((\sin(x))' = \lim_{\Delta x \to 0}\frac{\sin(x+\Delta x) - \sin(x)}{\Delta x} = \frac{\sin(x)\cos(\Delta x)+\cos(x)\sin(\Delta x)-\sin(x)}{\Delta x} = \frac{\cos(x)\sin(\Delta x)}{\Delta x} = \cos(x)\)
\((\cos(x))' = \lim_{\Delta x \to 0}\frac{\cos(x+\Delta x) - \cos(x)}{\Delta x} = \frac{\cos(x)\cos(\Delta x)-\sin(x)\sin(\Delta x)-\sin(x)}{\Delta x} = \frac{-\sin(x)\sin(\Delta x)}{\Delta x} = -\sin(x)\)
\((\tan(x))' = (\frac{\sin(x)}{\cos(x)})' = \frac{\cos^2(x)+\sin^2(x)}{\cos^2(x)} = \sec^2(x)\)