-
常数的导数
\((c)' = 0\) -
幂函数导数
\((x^a)' = a x^{a-1} \quad (a \in \mathbb{R})\) -
三角函数导数
\((\sin x)' = \cos x, \quad (\cos x)' = -\sin x\) -
正切、余切及正割、余割导数
\((\tan x)' = \sec^2 x, \quad (\cot x)' = -\csc^2 x\)
\((\sec x)' = \sec x \tan x, \quad (\csc x)' = -\csc x \cot x\) -
指数函数导数
\((a^x)' = a^x \ln a, \quad (e^x)' = e^x\) -
对数函数导数
\((\log_a x)' = \frac{1}{x \ln a}, \quad (\ln x)' = \frac{1}{x}\) -
反三角函数导数
\((\arcsin x)' = \frac{1}{\sqrt{1-x^2}}, \quad (\arccos x)' = -\frac{1}{\sqrt{1-x^2}}\)
\((\arctan x)' = \frac{1}{1+x^2}, \quad (\text{arccot } x)' = -\frac{1}{1+x^2}\) -
和差法则
\((u \pm v)' = u' \pm v'\) -
乘积与常数乘法法则
\((uv)' = u'v + uv', \quad (cu)' = c u' \quad (c \text{为常数})\) -
商法则与倒数法则
\(\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}, \quad \left( \frac{1}{v} \right)' = -\frac{v'}{v^2}\) -
反函数求导法则
\(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)
