【高等数学】常用极限、求导、级数

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常用极限，导数，级数
秒杀必背积分表实数部分
秒杀必背积分表三角部分

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#常用极限

$$\begin{gathered} \underset{x\to 0}{\lim }\frac{\sin x}{x}=1 \\ \underset{n\to \infty }{\lim }(1+\frac{1}{n}{)}^n=e;\underset{x\to \infty }{\lim }(1+\frac{1}{x}{)}^x=e \end{gathered}$$

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$$\begin{gathered} 当x\to 0: \\ \sin x\to x;\tan x\to x; \\ \arctan x\to x;\arcsin x\to x; \\ 1-\cos x\to \frac{x^2}{2};\sqrt[n]{1+x}\to \frac{1}{n}x; \\ {e}^x-1\to x;\ln (x+1)\to x; \\ (1+x{)}^{\alpha }\to \alpha x;\ln (x+\sqrt{1+x^2})\to x \\ {\log }_a(1+x)\to \frac{x}{\ln a};a^x-1\to x\ln a \\ x-\sin x\to \frac{1}{6}{x}^3;\tan x-x\to \frac{1}{3}{x}^3 \\ \arcsin x-x\to \frac{1}{6}{x}^3;x-\arctan x\to \frac{1}{3}{x}^3 \\ \tan x-\sin x\to \frac{1}{2}{x}^3 \\ 1-\sqrt{1-x^2}\to \frac{1}{2}{x}^2 \\ \sqrt{1+x^a}-1\to \frac{1}{2}{x}^a \end{gathered}$$

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$$\underset{x\to 0}{\lim }\frac{\sin \frac{1}{x}}{\frac{1}{x}}=0$$

#常用级数

$$\begin{gathered} \frac{1}{1-z}=\sum _{n=0}^{\infty }{z}^n=1+z+z^2\cdots +z^n+\cdots \\ \frac{1}{1+z}=\sum _{n=0}^{\infty }(-1{)}^n{z}^n=1-z+z^2\cdots +(-1{)}^n{z}^n+\cdots \\ \ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots +(-1{)}^n\frac{x^{n+1}}{n+1}+\cdots \\ {e}^x=1+x+\frac{x^2}{2}+\cdots +\frac{x^n}{n!}+\cdots \\ \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots +(-1{)}^{n+1}\frac{x^{2n-1}}{(2n-1)!}+\cdots \\ \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots +(-1{)}^{n+1}\frac{x^{2n-2}}{(2n-2)!}+\cdots \\ (1+x{)}^{\alpha }=1+\alpha x+\frac{\alpha (\alpha -1)}{2}{x}^2+\cdots \\ \sqrt{1+x^2}=1+\frac{1}{2}{x}^2-\frac{1}{8}{x}^4+o(x^4) \end{gathered}$$

#常用求导

$$\begin{gathered} (\arcsin x{)}^{\prime }=\frac{1}{\sqrt{1-x^2}} \\ (\arccos x{)}^{\prime }=-\frac{1}{\sqrt{1-x^2}} \\ (\arctan x{)}^{\prime }=\frac{1}{1+x^2} \\ (arccotx{)}^{\prime }=-\frac{1}{1+x^2} \\ (\tan x{)}^{\prime }=\frac{1}{{\cos }^{2}x}={\sec }^{2}x \\ (\cot x{)}^{\prime }=-\frac{1}{{\sin }^{2}x}=-{\csc }^{2}x \\ (\sec x{)}^{\prime }=\sec x\tan x=\frac{\tan x}{\cos x} \\ (\csc x{)}^{\prime }=-\csc x\cot x=-\frac{1}{\sin x\tan x} \\ (\cos x{)}^{(n)}=\cos (x+\frac{n\pi }{2}) \\ (\sin x{)}^{(n)}=\sin (x+\frac{n\pi }{2}) \\ (x^n{)}^{(n)}=n! \\ (x^n{)}^{(n+1)}=0 \\ (\frac{1}{x+a}{)}^{(n)}=\frac{(-1{)}^{n}n!}{(x+a{)}^{n+1}} \\ (\ln (x+b){)}^{(n)}=\frac{(-1{)}^{n-1}(n-1)!}{(x+b{)}^n} \end{gathered}$$

$$[f(x)\cdot g(x){]}^{(n)}=f^{(n)}g+C_n^1{f}^{(n-1)}g+\cdots +C_n^k{f}^{(n-k)}{g}^{(k)}+\cdots +f{g}^{(n)}$$

#求导的注意事项

$$对于函数，其输入的变量之间必须相互独立$$

$$\begin{gathered} 因此，对于以坐标为输入变量的多元函数，如 \\ (\rho ,\varphi ,z),(r,\theta ,\varphi ) \\ 其任意两个微商\frac{\partial \rho }{\partial \varphi }=0, \end{gathered}$$

全微分：