数学分析 CheatSheet

函数

三角横等式

$\tan^2x+1=\sec^2x$

$\sinh x = \frac{e^x}{2} - \frac{e^{-x}}{2}$

$\cosh x = \frac{e^x}{2} + \frac{e^{-x}}{2} = \sqrt{1+\sinh^2x}$

$sech(x) = \frac{2}{e^x + e^{-x}}$

$sin(x)cos(y) = \frac{sin(x+y) + sin(x-y)}{2}$

$sin(x)sin(y) = \frac{cos(x-y) - cos(x+y)}{2}$

$cos(x)cos(y) = \frac{cos(x-y) + cos(x+y)}{2}$

$sin(x) + sin(y) = 2sin(\frac{x+y}{2})cos(\frac{x-y}{2})$

$cos(x) + cos(y) = 2cos(\frac{x+y}{2})cos(\frac{x-y}{2})$

$cos(x) - cos(y) = -2sin(\frac{x+y}{2})cos(\frac{x-y}{2})$

微分

$\sin’x = \cos x$

$\cos’x = - \sin x$

$\tan’x = \sec^2x$

$\cot’x = - \csc^2x$

$\sec’x = \tan x \cdot \sec x$

$\csc’x = - \cot x \cdot \csc x$

$\cosh’ x = \sinh x$

$\sinh’ x = \cosh x$

$\tanh’ x = sech^2 x$

$arctanh(x)’ = \frac{1}{1-x^2}$

$arcsinh(x)’ = \frac{1}{\sqrt{1+x^2}}$

$arccosh(x)’ = \frac{1}{\sqrt{x-1}\sqrt{x+1}}$

有些地方^(-1)表示不是-1次方而是逆函数 $\tan^{-1} x = \arctan x$

$\arctan’x = \frac{1}{1+x^2}$

$arccot’x = -\frac{1}{1+x^2}$

$\arcsin’x = \frac{1}{\sqrt{1-x^2}}$