①常数，$y = c$，$y’ = c’ = 0$
②幂函数：$(x^a)’ = a \cdot x^{a - 1}$
$(\sqrt{x})’ = \frac{1}{2\sqrt{x}}$
$(\frac{1}{x})’ = -\frac{1}{x^2}$
$x^{-a} = \frac{1}{x^{a}}$
③指数函数 $(a^x)’ = a^x \cdot \ln a$
记: $(e^x)’ = e^x \cdot \ln e = e^x \cdot 1 = e^x$
$(e^{-x})’ = -e^{-x}$
$-\ln \frac{1}{x} = \ln x$
$\ln \frac{b}{a} = \ln b - \ln a$

③对数函数 $(\log_a x)’ = \frac{1}{x \ln a}$
记：$(\ln x)’ = (\log_e x)’ = \frac{1}{x \ln e} = \frac{1}{x}$
④三角函数
$\sec x = \frac{1}{\cos x}$

$\begin{cases} (\sec x)’ = \sec x \cdot \tan x \ (\csc x)’ = -\csc x \cdot \cot x \end{cases}$

$\begin{cases} (\sin x)’ = \cos x \ (\cos x)’ = -\sin x \end{cases}$

$\begin{cases} (\tan x)’ = \sec^2 x \ (\cot x)’ = -\csc^2 x \end{cases}$
注：含“c”加负号

⑤反三角函数
$\begin{cases} (\arcsin x)’ = \frac{1}{\sqrt{1 - x^2}} \ (\arccos x)’ = -\frac{1}{\sqrt{1 - x^2}} \end{cases}$

$\begin{cases} (\arctan x)’ = \frac{1}{1 + x^2} \ (\text{arccot} x)’ = -\frac{1}{1 + x^2} \end{cases}$

⑥ $\ln(x + \sqrt{x^2 \pm a})’ = \frac{1}{\sqrt{x^2 \pm a}}$，$\Rightarrow \ln(x + \sqrt{x^2 + 1})’ = \frac{1}{\sqrt{x^2 + 1}}$