# 常用公式集

# 三角函数常用公式

• $\Large \sin x = \frac{2\tan\frac{x}{2}}{1+\tan^2\frac{x}{2}}$

• $\Large \cos x = \frac{1-\tan^2\frac{x}{2}}{1+\tan^2\frac{x}{2}}$

• $\Large \tan x = \frac{2\tan\frac{x}{2}}{1-\tan^2\frac{x}{2}}$

• $\Large a\sin x + b\cos x = \sqrt{a^2+b^2}\sin(x+y), \tan y = \frac{b}{a}$

• $\Large \sin x = \sin(x+2k\pi)$

• $\Large \cos x = \cos(x+2k\pi)$

• $\Large \tan x = \tan(x+2k\pi)$

• $\Large \sin^2 x + \cos^2 x = 1$

• $\Large \sin(x+\pi) = -\sin x$

• $\Large \cos(x+\pi) = -\cos x$

• $\Large \tan(x+\pi) = \tan x$

• $\Large \sin(-x) = -\sin x$

• $\Large \cos(-x) = \cos x$

• $\Large \tan(-x) = -\tan x$

• $\Large \sin(\pi-x) = \sin x$

• $\Large \cos(\pi-x) = -\cos x$

• $\Large \tan(\pi-x) = -\tan x$

• $\Large \sin(\frac{\pi}{2}-x) = \cos x$

• $\Large \cos(\frac{\pi}{2}-x) = \sin x$

• $\Large \sin(\frac{\pi}{2}+x) = \cos x$

• $\Large \cos(\frac{\pi}{2}+x) = -\sin x$

• $\Large \cos(x-y) = \cos x \cos y + \sin x \sin y$

• $\Large \cos(x+y) = \cos x \cos y - \sin x \sin y$

• $\Large \sin(x+y) = \sin x \cos y \pm \cos x \sin y$

• $\Large \tan(x+y) = \frac{\tan x + \tan y}{1-\tan x \tan y}$

• $\Large \tan(x-y) = \frac{\tan x - \tan y}{1+\tan x \tan y}$

• $\Large \sin x + \sin y = 2\sin\frac{x+y}{2}\cos\frac{x-y}{2}$

• $\Large \sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2}$

• $\Large \cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2}$

• $\Large \cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2}$

• $\Large \tan x \pm \tan y = \frac{\sin(x \pm y)}{\cos x \cos y}$

• $\Large \cot x \pm \cot y = \pm\frac{\sin(x \pm y)}{\sin x \sin y}$

• $\Large \sin x \cos y = \frac{1}{2}[\sin(x+y)+\sin(x-y)]$

• $\Large \cos x \sin y = \frac{1}{2}[\sin(x+y)-\sin(x-y)]$

• $\Large \cos x \cos y = \frac{1}{2}[\cos(x+y)+\cos(x-y)]$

• $\Large \sin x \sin y = \frac{1}{2}[\cos(x+y)-\cos(x-y)]$

• $\Large \sin2x = 2\sin x\cos x$

• $\Large \cos2x = 2\cos^2x-1 = 1-2\sin^2x = \cos^2x-\sin^2x$

• $\Large \tan2x = \frac{2\tan x}{1-\tan^2x}$

# 双曲函数的7个常用公式

• $\Large \sh(x+y) = \sh x \ch y + \ch x \sh y$

• $\Large \sh(x-y) = \sh x \ch y - \ch x \sh y$

• $\Large \ch(x+y) = \ch x \ch y + \sh x \sh y$

• $\Large \ch(x-y) = \ch x \ch y - \sh x \sh y$

• $\Large \ch^2x-\sh^2x = 1$

• $\Large \sh 2x = 2\sh x \ch x$

• $\Large \ch 2x = \ch^2x+\sh^2x$

# 基本初等函数导数公式

• $\Large (C)'=0$

• $\Large (x^n)'=nx^{n-1}$

• $\Large (a^x)'=a^x\ln a$

• $\Large (e^x)'=e^x$

• $\Large (\log_a{x})'=\frac{1}{x\ln a}$

• $\Large (\ln x)' = \frac{1}{x}$

• $\Large (\sin x)'=\cos x$

• $\Large (\cos x)'=-\sin x$

• $\Large (\tan x)'=\sec^2 x$

• $\Large (\cot x)'=-\csc^2 x$

• $\Large (\sec x)'=\sec x\tan x$

• $\Large (\csc x)'=-\csc x\cot x$

• $\Large (\arcsin x)' =\frac{1}{\sqrt{1-x^2}}$

• $\Large (\arccos x)' =-\frac{1}{\sqrt{1-x^2}}$

• $\Large (\arctan x)' =\frac{1}{1+x^2}$

• $\Large (arc\cot x)' =-\frac{1}{1+x^2}$

• $\Large (\sh x)'=\ch x$

• $\Large (\ch x)'=\sh x$

• $\Large (\th x)'=\frac{1}{\ch^2x}$

• $\Large (ar\sh x)'=\frac{1}{\sqrt{1+x^2}}$

• $\Large (ar\ch x)'=\sqrt{x^2-1}$

• $\Large (ar\th x)'=\sqrt{1-x^2}$

# 常见函数高阶导数公式

• $\Large (e^x)^{(n)}=e^x$

• $\Large (\sin x)^{(n)}=\sin(x+\frac{n}{2}\pi)$

• $\Large (\cos x)^{(n)}=\cos(x+\frac{n}{2}\pi)$

• $\Large [\ln(1+x)]^{(n)}=(-1)^{n-1}\frac{(n-1)!}{(1+x)^n}$

• $\Large (x^{\mu})^{(n)}=\mu(\mu-1)(\mu-2)...(\mu-n+1)x^{\mu-n}$

• 当 $\mu=n$ 时，

$\Large (x^{\mu})^{(n)}=(x^n)^{(n)}=n!$