常见函数导数速查表
基本函数导数表
\[\begin{align} (1) & \quad (C)' = 0, \\ (2) & \quad (x^\mu)' = \mu x^{\mu-1}, \\ (3) & \quad (\sin x)' = \cos x, \\ (4) & \quad (\cos x)' = -\sin x, \\ (5) & \quad (\tan x)' = \sec^2 x, \\ (6) & \quad (\cot x)' = -\csc^2 x, \\ (7) & \quad (\sec x)' = \sec x \tan x, \\ (8) & \quad (\csc x)' = -\csc x \cot x, \\ (9) & \quad (a^x)' = a^x \ln a \quad (a > 0, a \neq 1), \\ (10) & \quad (e^x)' = e^x, \\ (11) & \quad (\log_a x)' = \frac{1}{x \ln a} \quad (a > 0, a \neq 1), \\ (12) & \quad (\ln x)' = \frac{1}{x}, \\ (13) & \quad (\arcsin x)' = \frac{1}{\sqrt{1 - x^2}}, \\ (14) & \quad (\arccos x)' = -\frac{1}{\sqrt{1 - x^2}}, \\ (15) & \quad (\arctan x)' = \frac{1}{1 + x^2}, \\ (16) & \quad (\operatorname{arccot} x)' = -\frac{1}{1 + x^2}. \end{align}\]函数导数定义
设函数f(x)在点x的某个邻域内有定义，如果函数f(x)在点x处的变化量与自变量x的变化量之比当x的变化量趋近于0时的极限存在，那么这个极限就称为函数f(x)在点x处的导数，记作f’(x)或df/dx。
\[\begin{equation} f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \end{equation}\]求导法则
\[\begin{align} \text{(和的求导法则)} & \quad (f(x) + g(x))' = f'(x) + g'(x), \\ \text{(差的求导法则)} & \quad (f(x) - g(x))' = f'(x) - g'(x), \\ \text{(积的求导法则)} & \quad (f(x) \cdot g(x))' = f'(x) \cdot g(x) + f(x) \cdot g'(x), \\ \text{(商的求导法则)} & \quad \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2}, \quad \text{假设 } g(x) \neq 0, \\ \text{(复合函数求导法则)} & \quad \left(f(g(x))\right)' = f'(g(x)) \cdot g'(x). \end{align}\]文档信息
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