导数与微分 | DaoDaoHome

导数的定义

f'(x_0) = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x}

f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}

f'(x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}

一、导数的四则运算法则

$(u \pm v)'=u' \pm v'$
$(uv)' = u'v+uv'$
$\left(\frac{u}{v}\right)' = \frac{u'v-uv'}{v^2}$

二、基本公式

$(c)' = 0$
$x^\mu = \mu x^{\mu-1}$
$(\sin x)' = \cos x$
$(\cos x)' = -\sin x$
$(\tan x)' = \sec^2 x$
$(\cot x)' = -\csc^2 x$
$(\sec x)' = \sec x \cdot \tan x$
$(\csc x)' = -\csc x \cdot \cot x$
$(e^x)' = e^x$
$(a^x)' = a^x \ln a$
$(\ln x)' = \frac{1}{x}$
$(\log_a x)' = \frac{1}{x \ln a}$
$(\arcsin x)' = \frac{1}{\sqrt{1-x^2}}$
$(\arccos x)' = -\frac{1}{\sqrt{1-x^2}}$
$(\arctan x)' = \frac{1}{1+x^2}$
$(\operatorname{arccot} x)' = -\frac{1}{1+x^2}$
$(x)' = 1$
$(\sqrt{x})' = \frac{1}{2\sqrt{x}}$

三、高阶导数的运算法则

$[u(x) \pm v(x)]^{(n)} = u^{(n)}(x) \pm v^{(n)}(x)$
$[cu(x)]^{(n)} = cu^{(n)}(x)$
$[u(ax+b)]^{(n)} = a^n u^{(n)}(ax+b)$
$[u(x) \cdot v(x)]^{(n)} = \sum_{k=0}^{n} C_n^k u^{(n-k)}(x) v^{(k)}(x)$

四、基本初等函数的n阶导数公式

$(x^n)^{(n)} = n!$
$(e^{ax+b})^{(n)} = a^n \cdot e^{ax+b}$
$(a^x)^{(n)} = a^x \ln^n a$
$[\sin(ax+b)]^{(n)} = a^n \sin \left(ax+b+n \cdot \frac{\pi}{2}\right)$
$[\cos(ax+b)]^{(n)} = a^n \cos \left(ax+b+n \cdot \frac{\pi}{2}\right)$
$\left(\frac{1}{ax+b}\right)^{(n)} = (-1)^n \frac{a^n \cdot n!}{(ax+b)^{n+1}}$
$[\ln(ax+b)]^{(n)} = (-1)^{n-1} \frac{a^n \cdot (n-1)!}{(ax+b)^n}$

五、微积分运算法则

$d(u \pm v) = du \pm dv$
$d(cu) = cdu$
$d(uv) = vdu + udv$
$d\left(\frac{u}{v}\right) = \frac{vdu - udv}{v^2}$

六、常用微分公式

$d(c) = 0$
$d(x^\mu) = \mu x^{\mu-1}dx$
$d (\sin x) = \cos xdx$
$d (cos x) = - sin xdx$
$d (tan x) = \sec^2 xdx$
$d (\cot x) = -\csc^2 xdx$
$\int sec xdx = \ln |\sec x + \tan x| + c$
$d (\csc x) = -\csc x\cdot \cot xdx$
$d(e^x) = e^x dx$
$d (a^x) = a^x \ln adx$
$d (\ln x) = \frac{1}{x}dx$
$d (\log_a x) = \frac{1}{x \ln a}dx$
$d (\arcsin x) = \frac{1}{\sqrt{1-x^2}}dx$
$d (\arccos x) = -\frac{1}{\sqrt{1-x^2}}dx$
$d (\arctan x) = \frac{1}{1+x^2}dx$
$d (\operatorname{arc} \cot x) = -\frac{1}{1+x^2}dx$

七、基本积分公式

$\int kdx = kx+c$
$\int x^\mu dx = \frac{x^{\mu+1}}{\mu+1}+c$
$\int \frac{dx}{x} = \ln |x|+c$
$\int a^x dx = \frac{a^x}{\ln a}+c$
$\int \cos xdx = \sin x+c$
$\int \sin xdx = -\cos x+c$
$\int \frac{1}{\cos^2 x}dx = \int \sec^2 xdx = \tan x + c$
$\int \frac{1}{\sin^2 x}csc^2xdx = -\cot x+c$
$\int \frac{1}{1+x^2}dx=\arctan x+c$
$\int \frac{1}{\sqrt{1-x^2}} dx = \arcsin x+c$

八、补充积分公式

$\int \tan xdx=-\ln |\cos x|+c$
$\int \cot xdx = \ln |\sin x| + c$
$\int \csc xdx = \ln |\csc x-\cot x| + c$
$\int \frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan\frac{x}{a}+c$
$\int \frac{1}{x^2-a^2}dx=\frac{1}{2a} \ln |\frac{x-a}{x+a}|+c$
$\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin\frac{x}{a}+c$
$\int \frac{1}{\sqrt{x^2\pm a^2}}dx=\ln|x+\sqrt{x^2\pm a^2}|+c$

九、常用凑微分公式

积分型	换元公式
$f(ax+b)dx = \frac{1}{a}\int f(ax+b)d(ax+b)$	$u=ax+b$
$f(x^{\mu})x^{\mu-1}dx = \frac{1}{\mu}\int f(x^{\mu})d(x^{\mu})$	$u = x^n$
$f(\ln x) \cdot \frac{1}{x} dx = \int f(\ln x) d(\ln x)$	$u=\ln x$
$f(e^{x}) \cdot e^{x} dx = \int f(e^{x}) d(e^{x})$	$u=e^x$
$f(a^{x}) \cdot a^{x} dx = \frac{1}{\ln a} \int f(a^{x}) d(a^{x})$	$u=a^x$
$f(\sin x) \cdot \cos x dx = \int f(\sin x) d(\sin x)$	$u = \sin x$
$f(\cos x) \cdot \sin x dx = -\int f(\cos x) d(\cos x)$	$u = \cos x$
$f(\tan x) \cdot \sec^{2} x dx = \int f(\tan x) d(\tan x)$	$u = \tan x$
$f(\cot x) \cdot \csc^{2} x dx = \int f(\cot x) d(\cot x)$	$u = \cot x$
$f(\arctan x) \cdot \frac{1}{1+x^{2}} dx = \int f(\arctan x) d(\arctan x)$	$u = \arctan x$
$f(\arcsin x) \cdot \frac{1}{\sqrt{1-x^{2}}} dx = \int f(\arcsin x) d(\arcsin x)$	$u = \arcsin x$

十、分部积分法公式

(1) 形如： $x^{n}e^{ax}dx, u=x^{n}, dv=e^{ax}dx$

形如： $x^{n} \sin xdx \quad u=x^{n}, dv = \sin xdx$

形如： $x^{n} \cos xdx \quad u=x^{n}, dv = \cos xdx$

(2) 形如： $x^{n} \arctan xdx, \quad u=\arctan x, \quad dv=x^{n}dx$

形如： $x^{n} \ln xdx, \quad u=\ln x, \quad dv=x^{n}dx$

(3) 形如： $e^{ax} \sin xdx, \int e^{ax} \cos xdx \quad u=e^{ax}, \sin x, \cos x$

十一、第二换元积分法中的三角换元公式

$\sqrt{a^{2}-x^{2}} \quad x = a \sin t$
$\sqrt{a^{2}+x^{2}} \quad x = a \tan t$
$\sqrt{x^{2}-a^{2}} \quad x = a \sec t$

【特殊角的三角函数值】

$\sin0=0$
$\sin\frac{\pi}{6}=\frac{1}{2}$
$\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$
$\sin\frac{\pi}{2}=1$
$\sin\pi=0$
$\cos0=1$
$\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$
$\cos\frac{\pi}{3}=\frac{1}{2}$
$\cos\frac{\pi}{2}=0$
$\cos\pi=-1$
$\tan0=0$
$\tan\frac{\pi}{6}=\frac{\sqrt{3}}{3}$
$\tan\frac{\pi}{3}=\sqrt{3}$
$\tan\frac{\pi}{2}$ 不存在
$\tan\pi=0$
$\cot0$ 不存在
$\cot\frac{\pi}{6}=\sqrt{3}$
$\cot\frac{\pi}{3}=\frac{\sqrt{3}}{3}$
$\cot\frac{\pi}{2}=0$
$\cot\pi$ 不存在

十二、重要公式

$\lim_{x\to0}\frac{\sin x}{x}=1$
$\lim_{x\to0}(1+x)^{\frac{1}{x}}=e$
$\lim_{n\to\infty}\sqrt[n]{a}(a>0)=1$
$\lim_{n\to\infty}\sqrt[n]{n}=1$
$\lim_{x\to+\infty}\arctan x=\frac{\pi}{2}$
$\lim_{x\to-\infty}\arctan x=-\frac{\pi}{2}$
$\lim_{x\to+\infty}\operatorname{arccot}x=0$
$\lim_{x\to-\infty}\operatorname{arccot}x=\pi$
$\lim_{x\to0}e^{\frac{1}{x}}=0$
$\lim_{x\to0}e^{\frac{1}{x}}=\infty$
$\lim_{x\to0}x^{x}=1$
$\lim_{x\to\infty}\frac{a_0x^n+a_1x^{n-1}+\dots+a_n}{b_0x^m+b_1x^{m-1}+\dots+b_m}=\begin{cases}\frac{a_0}{b_0} & n=m \\ 0 & n<m \\ \infty & n>m\end{cases}$ (系数不为0的情况)

十三、常用等价无穷小关系(X->0)

$\sin x-x$
$\tan x-x$
$\arcsin x - x$
$\arctan x - x$
$1-\cos x-\frac{1}{2}x^2$
$\ln(1+x)-x$
$e^{x}-1-x$
$a^{x}-1-x\ln a$
$(1+x)^{x}-1-ex$

十四、三角函数公式

1.两角和公式

$\sin(A+B) = \sin A \cos B + \cos A \sin B$
$\cos(A+B) = \cos A \cos B - \sin A \sin B$
$\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
$\sin(A-B) = \sin A \cos B - \cos A \sin B$
$\cos(A-B) = \cos A \cos B + \sin A \sin B$
$\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$
$\cot(A+B) = \frac{\cot A \cdot \cot B - 1}{\cot B + \cot A}$
$\cot(A-B) = \frac{\cot A \cdot \cot B + 1}{\cot B - \cot A}$

2.二倍角公式

$\sin 2A = 2 \sin A \cos A$
$\cos 2A = \cos^2 A - \sin^2 A = 1 - 2 \sin^2 A = 2 \cos^2 A - 1$
$\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}$

3.半角公式

$\sin \frac{A}{2} = \sqrt{\frac{1 - \cos A}{2}}$
$\cos \frac{A}{2} = \sqrt{\frac{1 + \cos A}{2}}$
$\tan \frac{A}{2} = \sqrt{\frac{1 - \cos A}{1 + \cos A}} = \frac{\sin A}{1 + \cos A}$
$\cot \frac{A}{2} = \sqrt{\frac{1 + \cos A}{1 - \cos A}} = \frac{\sin A}{1 - \cos A}$

4. 和差化积公式

$\sin a + \sin b = 2 \sin \frac{a+b}{2} \cdot \cos \frac{a-b}{2}$
$\sin a - \sin b = 2 \cos \frac{a+b}{2} \cdot \sin \frac{a-b}{2}$
$\cos a + \cos b = 2 \cos \frac{a+b}{2} \cdot \cos \frac{a-b}{2}$
$\cos a - \cos b = -2 \sin \frac{a+b}{2} \cdot \sin \frac{a-b}{2}$
$\tan a + \tan b = \frac{\sin(a+b)}{\cos a \cdot \cos b}$

5. 积化和差公式

$\sin a \sin b = -\frac{1}{2}[\cos(a+b) - \cos(a-b)]$
$\cos a \cos b = \frac{1}{2}[\cos(a+b) + \cos(a-b)]$
$\sin a \cos b = \frac{1}{2}[\sin(a+b) + \sin(a-b)]$
$\cos a \sin b = \frac{1}{2}[\sin(a+b) - \sin(a-b)]$

6. 万能公式

$\sin a = \frac{2 \tan \frac{a}{2}}{1 + \tan^2 \frac{a}{2}}$
$\cos a = \frac{1 - \tan^2 \frac{a}{2}}{1 + \tan^2 \frac{a}{2}}$
$\tan a = \frac{2 \tan \frac{a}{2}}{1 - \tan^2 \frac{a}{2}}$

7. 平方关系

$\sin^2 x + \cos^2 x = 1$
$\sec^2 x - \tan^2 x = 1$
$\csc^2 x - \cot^2 x = 1$

8. 倒数关系

$\tan x \cdot \cot x = 1$
$\sec x \cdot \cos x = 1$
$\csc x \cdot \sin x = 1$

9. 商数关系

$\tan x = \frac{\sin x}{\cos x}$
$\cot x = \frac{\cos x}{\sin x}$

几种常见的微分方程

可分离变量的微分方程： $\frac{dy}{dx} = f(x)g(y), \quad f_1(x)g_1(y)dx + f_2(x)g_2(y)dy = 0$
齐次微分方程: $\frac{dy}{dx} = f(\frac{y}{x})$