考研必备数学公式大全（持续更新）

基础回顾

面（体）积公式

$$\begin{array}{rl}& \\ & 球表面积公式：S=4\pi R^2\\ & \\ & 球体积公式：V=\frac{4}{3}\pi R^3\\ & \\ & 圆锥体积公式：V=\frac{1}{3}sh(s为圆锥底面积，h为圆锥的高)\\ & \\ & 椭圆面积公式：S=\pi ab\\ & \\ & 扇形面积公式：S=\frac{1}{2}lr=\frac{1}{2}{r}^2\theta （其中l为弧长，r为半径，\theta 为夹角(用\pi 表示)）\end{array}$$

一元二次方程基础

$$\begin{array}{rl}& \\ & 一元二次方程：a{x}^2+bx+c=0(a\ne 0)\\ & \\ & 根的公式x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ & \\ & 韦达定理：x_1+x_2=-\frac{b}{a}{x}_1{x}_2=\frac{c}{a}\\ & \\ & 判别式：\Delta =b^2-4ac⟹\{\begin{array}{l}\Delta >0，两个不等实根\\ \Delta =0，两个相等实根\\ \Delta <0，两个共轭的复根（无实根）\end{array}\\ & \\ & 抛物线y=a{x}^2+bx+c的顶点：(-\frac{b}{2a},c-\frac{b^2}{4a})\end{array}$$

极坐标方程与直角坐标转换

$$\begin{array}{rl}& 直角坐标化极坐标\{\begin{array}{l}x=\rho \cos \theta \\ y=\rho \sin \theta \end{array}⟹x^2+y^2={\rho }^2\\ & \\ & 极坐标化直角坐标：{\rho }^2=x^2+y^2⟹\tan \theta =\frac{y}{x}\\ & \end{array}$$

切线与法线方程

$$\begin{array}{rl}& 切线方程：\frac{y-y_0}{x-x_0}=f^{\prime }(x_0)，即y-y_0=f^{\prime }(x_0)(x-x_0)\\ & \\ & 法线方程：\frac{y-y_0}{x-x_0}=-\frac{1}{f^{\prime }(x_0)},即y-y_0=-\frac{1}{f^{\prime }(x_0)}(x-x_0)\end{array}$$

因式分解公式

$$\begin{array}{rl}& \\ & (a+b{)}^2=a^2+2ab+b^2\\ & \\ & (a-b{)}^2=a^2-2ab+b^2\\ & \\ & (a+b+c{)}^2=a^2+b^2+c^2+2ab+2ac+2bc\\ & \\ & (a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3\\ & \\ & (a-b{)}^3=a^3-3a^{2}b+3a{b}^2-b^3\\ & \\ & (a+b)(a-b)=a^2-b^2\\ & \\ & a^3+b^3=(a+b)(a^2-ab+b^2)\\ & \\ & a^3-b^3=(a-b)(a^2+ab+b^2)\\ & \\ & a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots +a{b}^{n-2}+b^{n-1})\\ & \\ & (a+b{)}^n=\sum _{k=0}^n{C}_n^k{a}^k{b}^{n-k}=a^n+n{a}^{n-1}b+\frac{n(n-1)}{2!}{a}^{n-1}{b}^2+\cdots +\frac{n(n-1)\cdots (n-k+1)}{k!}{a}^{n-k}{b}^k+\cdots +na{b}^{n-1}+b^n\end{array}$$

阶乘与双阶乘

n ! = 1 × 2 × 3 × . . . × n       ( 规 定 0 ! = 1 ) ( 2 n ) ! ! = 2 × 4 × 6 × . . . × ( 2 n ) = 2 n ⋅ n ! ( 2 n − 1 ) ! ! = 1 × 3 × 5... × ( 2 n − 1 ) \begin{aligned} & n! = 1\times2\times3\times ... \times n ~~~~~(规定0!=1) \\ \\ & (2n)!! = 2\times4\times6\times ... \times (2n) = 2^n \cdot n! \\ \\ & (2n-1)!! = 1\times3\times5...\times(2n-1) \end{aligned} ​n!=1×2×3×...×n     (规定0!=1)(2n)!!=2×4×6×...×(2n)=2n⋅n!(2n−1)!!=1×3×5...×(2n−1)​

函数的奇偶性

$$\begin{array}{rl}& 定义在[-a,a]上的任一函数，可以表示为一个奇函数与一个偶函数之和：\\ & \\ & f(x)=\frac{1}{2}[f(x)-f(-x)]+\frac{1}{2}[f(x)+f(-x)]\end{array}$$

排列组合

$$\begin{array}{rl}{A}_n^m& =n(n-1)(n-2)\cdots (n-m+1)\\ & \\ & =\frac{n!}{(n-m)!}\\ & \\ & \\ C_n^m& =\frac{A_n^m}{m!}=\frac{n(n-1)\cdots (n-m+1)}{m!}\\ & \\ & =\frac{n!}{m!(n-m)!}\end{array}$$

等差数列

$$\begin{array}{rl}& a_n=a_1+(n-1)d\\ & \\ & S_n=n{a}_1+\frac{n(n-1)}{2}dn\in N^{\ast }\\ & \\ & S_n=\frac{n(a_1+a_n)}{2}\\ & \end{array}$$

等比数列

$$\begin{array}{rl}& a_n=a_1\cdot q^{n-1}\\ & \\ & S_n=\frac{a_1(1-q^n)}{1-q}(q\ne 1)\end{array}$$

常用数列前n项和

$$\begin{array}{rl}& \\ & \sum _{k=1}^{n}k=1+2+3+\cdots +n=\frac{n(n+1)}{2}\\ & \\ & \sum _{k=1}^n(2k-1)=1+3+5+\cdots +(2n-1)=n^2\\ & \\ & \sum _{k=1}^n{k}^2=1^2+2^2+3^2+\cdots +n^2=\frac{n(n+1)(2n+1)}{6}\\ & \\ & \sum _{k=1}^n{k}^3=1^3+2^3+3^3+\cdots +n^3=[\frac{n(n+1)}{2}{]}^2=(\sum _{k=1}^{n}k{)}^2\\ & \\ & \sum _{k=1}^{n}k(k+1)=1\times 2+2\times 3+3\times 4+\cdots +n(n+1)=\frac{n(n+1)(n+2)}{3}\\ & \\ & \sum _{k=1}^n\frac{1}{k(k+1)}=\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\cdots +\frac{1}{n(n+1)}=\frac{n}{n+1}\end{array}$$

不等式

$$\begin{array}{rl}& \\ & 2|ab|\le a^2+b^2\\ & \\ & |a\pm b|\le |a|+|b|\\ & \\ & ||a|-|b||\le |a-b|\\ & \\ & |a_1\pm a_2\pm \cdot \cdot \cdot \cdot \pm a_n|\le |a_1|+|a_2|+\cdot \cdot \cdot +|a_n|\\ & \\ & |{\int }_a^{b}f(x)dx|\le {\int }_a^b|f(x)|dx(a<b)\\ & \\ & \sqrt{ab}\le \frac{a+b}{2}\le \sqrt{\frac{a^2+b^2}{2}}(a,b>0)\\ & \\ & \sqrt[3]{abc}\le \frac{a+b+c}{3}\le \sqrt{\frac{a^2+b^2+c^2}{3}}(a,b,c>0)\\ & \\ & \sqrt[n]{a_1{a}_2\cdot \cdot \cdot a_n}\le \frac{a_1+a_2+...+a_n}{n}\le \sqrt{\frac{{a_1}^2+{a_2}^2+...+{a_n}^2}{n}}（a_1,a_2,...a_n>0，等号当且仅当a_1=a_2=...=a_n时成立）\\ & \\ & xy\le \frac{x^p}{p}+\frac{x^q}{q}(x,y,p,q>0,\frac{1}{p}+\frac{1}{q}=1)\\ & \\ & (ac+bd{)}^2\le (a^2+b^2)(c^2+d^2)\\ & \\ & (a_1{b}_1+a_2{b}_2+a_3{b}_3{)}^2\le ({a_1}^2+{a_2}^2+{a_3}^2)({b_1}^2+{b_2}^2+{b_3}^2)\\ & \\ & [{\int }_a^{b}f(x)\cdot g(x)dx{]}^2\le {\int }_a^b{f}^2(x)dx\cdot {\int }_a^b{g}^2(x)dx\\ & \\ & \sin x<x<\tan x(0<x<\frac{\pi }{2})\\ & \\ & \arctan x\le x\le \arcsin x(0\le x\le 1)\\ & \\ & x+1\le e^x\\ & \\ & \ln x\le x-1\\ & \\ & \frac{1}{1+x}<\ln (1+\frac{1}{x})<\frac{1}{x}(x>0)\end{array}$$

三角函数公式

诱导公式

$$\begin{array}{rl}& \sin (-\alpha )=-\sin \alpha \\ & \\ & \cos (-\alpha )=\cos \alpha \\ & \\ & \sin (\frac{\pi }{2}-\alpha )=\cos \alpha \\ & \\ & \cos (\frac{\pi }{2}-\alpha )=\sin \alpha \\ & \\ & \sin (\frac{\pi }{2}+\alpha )=\cos \alpha \\ & \\ & \cos (\frac{\pi }{2}+\alpha )=-\sin \alpha \\ & \\ & \sin (\pi -\alpha )=\sin \alpha \\ & \\ & \cos (\pi -\alpha )=-\cos \alpha \\ & \\ & \sin (\pi +\alpha )=-\sin \alpha \\ & \\ & \cos (\pi +\alpha )=-\cos \alpha \\ & \\ & \\ & 奇变偶不变，符号看象限\\ & 奇指k\cdot \frac{\pi }{2}中k\\ & 看象限是指：将\alpha 看成锐角，然后看{\sin }_{{}_{(变之前的，或cos)}}(k\cdot \frac{\pi }{2}\pm \alpha )的符号\end{array}$$

平方关系

$$\begin{array}{rl}& \\ & 1+{\tan }^2\alpha ={\sec }^2\alpha \\ & \\ & 1+{\cot }^2\alpha ={\csc }^2\alpha \\ & \\ & {\sin }^2\alpha +{\cos }^2\alpha =1\end{array}$$

两角和与差的三角函数

$$\begin{array}{rl}& \sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\ & \\ & \cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\ & \\ & \sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\ & \\ & \cos (\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \cos \beta \\ & \\ & \tan (\alpha +\beta )=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }\\ & \\ & \tan (\alpha -\beta )=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }\end{array}$$

积化和差公式

$$\begin{array}{rl}& \cos \alpha \cos \beta =\frac{1}{2}[\cos (\alpha +\beta )+cos(\alpha -\beta )]\\ & \\ & \cos \alpha \sin \beta =\frac{1}{2}[\sin (\alpha +\beta )-\sin (\alpha -\beta )]\\ & \\ & \sin \alpha \cos \beta =\frac{1}{2}[\sin (\alpha +\beta )+\sin (\alpha -\beta )]\\ & \\ & \sin \alpha \sin \beta =-\frac{1}{2}[\cos (\alpha +\beta )-\cos (\alpha -\beta )]\end{array}$$
记忆口诀：
积化和差得和差，余弦在后要想加。异名函数取正弦，正弦相乘取负号。

和差化积公式

$$\begin{array}{rl}& \sin \alpha +\sin \beta =2\sin \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}\\ & \\ & \sin \alpha -\sin \beta =2\cos \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}\\ & \\ & \cos \alpha +\cos \beta =2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}\\ & \\ & \cos \alpha -\cos \beta =-2\sin \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}\end{array}$$
记忆口诀：
正加正，正在前，正减正，余在前；余加余，余并肩；余减余，负正弦

倍角公式

$$\begin{array}{rl}& \sin 2\alpha =2\sin \alpha \cos \alpha \\ & \\ & \cos 2\alpha ={\cos }^2\alpha -{\sin }^2\alpha =1-2{\sin }^2\alpha =2{\cos }^2\alpha -1\\ & \\ & \sin 3\alpha =-4{\sin }^3\alpha +3\sin \alpha \\ & \\ & \cos 3\alpha =4{\cos }^3\alpha -3\cos \alpha \\ & \\ & {\sin }^2\alpha =\frac{1-\cos 2\alpha }{2}\\ & \\ & {\cos }^2\alpha =\frac{1+\cos 2\alpha }{2}\\ & \\ & \tan 2\alpha =\frac{2\tan \alpha }{1-{\tan }^2\alpha }\\ & \\ & \cot 2\alpha =\frac{{\cot }^2\alpha -1}{2\cot \alpha }\end{array}$$

半角公式

$$\begin{array}{rl}& {\sin }^2\frac{\alpha }{2}=\frac{1-\cos \alpha }{2}\\ & \\ & {\cos }^2\frac{\alpha }{2}=\frac{1+\cos \alpha }{2}\\ & \\ & \sin \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{2}}\\ & \\ & \cos \frac{\alpha }{2}=\pm \sqrt{\frac{1+\cos \alpha }{2}}\\ & \\ & \tan \frac{\alpha }{2}=\frac{1-\cos \alpha }{\sin \alpha }=\frac{\sin \alpha }{1+\cos \alpha }=\pm \sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }}\\ & \\ & \cot \frac{\alpha }{2}=\frac{\sin \alpha }{1-\cos \alpha }=\frac{1+\cos \alpha }{\sin \alpha }=\pm \sqrt{\frac{1+\cos \alpha }{1-\cos \alpha }}\end{array}$$

万能公式

$$\begin{array}{rl}& \sin \alpha =\frac{2\tan \frac{\alpha }{2}}{1+{\tan }^2\frac{\alpha }{2}}\\ & \\ & \cos \alpha =\frac{1-{\tan }^2\frac{\alpha }{2}}{1+{\tan }^2\frac{\alpha }{2}}\end{array}$$

其他公式

$$\begin{array}{rl}& 1+\sin \alpha =(\sin \frac{\alpha }{2}+\cos \frac{\alpha }{2}{)}^2\\ & \\ & 1-\sin \alpha =(\sin \frac{\alpha }{2}-\cos \frac{\alpha }{2}{)}^2\end{array}$$

反三角函数恒等式

$$\begin{array}{rl}& \arcsin x+\arccos x=\frac{\pi }{2}\\ & \\ & \arctan x+arccotx=\frac{\pi }{2}\\ & \\ & \sin (\arccos x)=\sqrt{1-x^2}\\ & \\ & \cos (\arcsin x)=\sqrt{1-x^2}\\ & \\ & \sin (\arcsin x)=x\\ & \\ & \arcsin (\sin x)=x\\ & \\ & \cos (\arccos x)=x\\ & \\ & \arccos (\cos x)=x\\ & \\ & \arccos (-x)=\pi -\arccos x\end{array}$$

极限相关公式

数列极限递推式

$$\begin{array}{rl}& a_{n+1}=f(a_n)\\ & \\ 结论一：& f^{\prime }(x)>0，\{\begin{array}{l}{a}_2>a_1⟹\{a_n\}\nearrow 单调递增\\ a_2<a_1⟹\{a_n\}\searrow 单调递减\end{array}\\ & \\ 结论二（压缩映像原理）：& \exists k\in (0,1)，使得|f^{\prime }(x)|\le k⟹a_n收敛\end{array}$$

重要极限公式

$$\begin{array}{rl}& \underset{x\to 0^{+}}{\lim }{x}^{\alpha }\ln x=0其中\alpha >0\\ & \\ & \underset{x\to 0^{+}}{\lim }{x}^{\alpha }(\ln x{)}^k=0其中\alpha >0，k>0\\ & \\ & \underset{x\to +\infty }{\lim }{x}^{\alpha }{e}^{-\delta x}=0其中\alpha >0，\delta >0\\ & \\ & \underset{x\to 0}{\lim }\frac{\sin x}{x}=1⟹\underset{\varphi (x)\to 0}{\lim }\frac{\sin \varphi (x)}{\varphi (x)}=1其中\varphi (x)\ne 0\\ & \\ & \underset{x\to 0}{\lim }(1+x{)}^{\frac{1}{x}}=e⟹\underset{\varphi (x)\to 0}{\lim }(1+\varphi (x){)}^{\frac{1}{\varphi (x)}}=e其中\varphi (x)\ne 0\\ & \\ & \underset{n\to \infty }{\lim }\sqrt[n]{n}=1\\ & \\ & \underset{n\to \infty }{\lim }\sqrt[n]{a}=1（常数a>0）\\ & \end{array}$$

常用等价无穷小

$$\begin{array}{rl}& x\to 0时，\\ & \\ & \sin x\sim \tan x\sim \arcsin x\sim \arctan x\sim (e^x-1)\sim \ln (1+x)\sim x,1-\cos x\sim \frac{1}{2}{x}^2,\\ & \\ & (1+x{)}^a-1\sim ax,a^x-1\sim x\ln a(a>0,a\ne 1)\end{array}$$

1^∞ 型

$$\lim u^v=e^{\lim (u-1)v}(其中\lim u=1,\lim v=\infty ，即1^{\infty }型)$$

导数相关公式

导数定义

$$\begin{array}{rl}& f^{\prime }(x_0)=\underset{\Delta x\to 0}{\lim }\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}\\ & \\ & f^{\prime }(x_0)=\underset{x\to x_0}{\lim }\frac{f(x)-f(x_0)}{x-x_0}\end{array}$$

微分定义

Δ y = f ( x 0 + Δ x ) − f ( x 0 ) Δ y = A Δ x + o ( Δ x ) A Δ x = f ′ ( x 0 ) Δ x \begin{aligned} & \Delta y = f(x_0 + \Delta x) - f(x_0) \\ \\ & \Delta y = A\Delta x + o(\Delta x) \\ \\ & A\Delta x = f'(x_0) \Delta x \end{aligned} ​Δy=f(x0​+Δx)−f(x0​)Δy=AΔx+o(Δx)AΔx=f′(x0​)Δx​

连续，可导及可微关系

一元函数

可微

连续

可导

多元函数

一阶偏导数连续

可微

连续

可导

导数四则运算

$$\begin{array}{rl}& [u(x)\pm v(x){]}^{\prime }=u^{\prime }(x)\pm v^{\prime }(x)\\ & \\ & [u(x)v(x){]}^{\prime }=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)\\ & \\ & [u(x)v(x)w(x){]}^{\prime }=u^{\prime }(x)v(x)w(x)+u(x)v^{\prime }(x)w(x)+u(x)v(x)w^{\prime }(x)\\ & \\ & {\left[\begin{array}{c}\frac{u(x)}{v(x)}\end{array}\right]}^{\prime }=\frac{u^{\prime }(x)v(x)-u(x)v^{\prime }(x)}{[v(x){]}^2}\\ & \end{array}$$

复合函数求导

$$\{f[g(x)]{\}}^{\prime }=f^{\prime }[g(x)]g^{\prime }(x)$$

反函数求导

$$\begin{array}{rl}& y=f(x),x=\phi (y)⟹{\phi }^{\prime }(y)=\frac{1}{f^{\prime }(x)}\\ & \\ & y_x^{\prime }=\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}=\frac{1}{x_y^{\prime }}\\ & \\ & y_{xx}^{{}^{\prime \prime }}=\frac{d^{2}y}{d{x}^2}=\frac{d(\frac{dy}{dx})}{dx}=\frac{d(\frac{1}{x_y^{\prime }})}{dx}=\frac{d(\frac{1}{x_y^{\prime }})}{dy}\cdot \frac{dy}{dx}=\frac{d(\frac{1}{x_y^{\prime }})}{dy}\cdot \frac{1}{x_y^{\prime }}=\frac{-x_{yy}^{{}^{\prime \prime }}}{(x_y^{\prime }{)}^3}\end{array}$$

参数方程求导

$$\begin{array}{rl}& \{\begin{array}{l}x=\phi (t)\\ y=\psi (t)\end{array}\\ & \\ & \frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{{\psi }^{\prime }(t)}{{\phi }^{\prime }(t)}\\ & \\ & \frac{d^{2}y}{d{x}^2}=\frac{d(\frac{dy}{dx})}{dx}=\frac{d(\frac{dy}{dx})/dt}{dx/dt}=\frac{{\psi }^{\prime \prime }(t){\phi }^{\prime }(t)-{\psi }^{\prime }(t){\phi }^{\prime \prime }(t)}{[{\phi }^{\prime }(t){]}^3}\end{array}$$

变限积分求导公式

$$\begin{array}{rl}& 设F(x)={\int }_{{\phi }_1(x)}^{{\phi }_2(x)}f(t)dt,则\\ & \\ & F^{\prime }(x)=\frac{d}{dx}\left[\begin{array}{c}{\int }_{{\phi }_1(x)}^{{\phi }_2(x)}f(t)dt\end{array}\right]=f[{\phi }_2(x)]{\phi }_2^{\prime }(x)-f[{\phi }_1(x)]{\phi }_1^{\prime }(x)\end{array}$$

基本初等函数的导数公式（❤❤❤）

$$\begin{array}{rl}& (x^a{)}^{\prime }=a{x}^{a-1}(a为常数)\\ & \\ & (a^x{)}^{\prime }=a^x\ln a\\ & \\ & (e^x{)}^{\prime }=e^x\\ & \\ & (lo{g}_{a}x{)}^{\prime }=\frac{1}{x\ln a}(a>0,a\ne 1)\\ & \\ & (\ln x{)}^{\prime }=\frac{1}{x}\\ & \\ & (\sin x{)}^{\prime }=\cos x\\ & \\ & (\cos x{)}^{\prime }=-\sin x\\ & \\ & (\arcsin x{)}^{\prime }=\frac{1}{\sqrt{1-x^2}}\\ & \\ & (\arccos x{)}^{\prime }=-\frac{1}{\sqrt{1-x^2}}\\ & \\ & (\tan x{)}^{\prime }={\sec }^{2}x\\ & \\ & (\cot x{)}^{\prime }=-{\csc }^{2}x\\ & \\ & (\arctan x{)}^{\prime }=\frac{1}{1+x^2}\\ & \\ & (arccotx{)}^{\prime }=-\frac{1}{1+x^2}\\ & \\ & (\sec x{)}^{\prime }=\sec x\cdot \tan x\\ & \\ & (\csc x{)}^{\prime }=-\csc x\cdot \cot x\\ & \\ & [\ln (x+\sqrt{x^2+1}){]}^{\prime }=\frac{1}{\sqrt{x^2+1}}\\ & \\ & [\ln (x+\sqrt{x^2-1}){]}^{\prime }=\frac{1}{\sqrt{x^2-1}}\\ & \end{array}$$

高阶导数的运算

$$[u\pm v{]}^{(n)}=u^{(n)}\pm v^{(n)}$$
$$\begin{array}{rl}(uv{)}^{(n)}& =u^{(n)}v+C_n^1{u}^{(n-1)}{v}^{\prime }+C_n^2{u}^{(n-2)}{v}^{\prime \prime }+...+C_n^k{u}^{(n-k)}{v}^{(k)}+...+C_n^{n-1}{u}^{\prime }{v}^{(n-1)}+u{v}^{(n)}\\ & =\sum _{k=0}^n{C}_n^k{u}^{(n-k)}{v}^{(k)}\end{array}$$

常用初等函数的n阶导数公式

$$\begin{array}{rl}& (a^x{)}^{(n)}=a^x(\ln a{)}^n\\ & \\ & (e^x{)}^{(n)}=e^x\\ & \\ & (\sin kx{)}^{(n)}=k^n\sin (kx+n\cdot \frac{\pi }{2})\\ & \\ & (\cos kx{)}^{(n)}=k^n\cos (kx+n\cdot \frac{\pi }{2})\\ & \\ & (\ln x{)}^{(n)}=(-1{)}^{n-1}\frac{(n-1)!}{x^n}(x>0)\\ & \\ & [\ln (1+x){]}^{(n)}=(-1{)}^{n-1}\frac{(n-1)!}{(x+1{)}^n}(x>-1)\\ & \\ & [(x+x_0{)}^m{]}^{(n)}=m(m-1)(m-2)\cdot \cdot \cdot \cdot (m-\\ & \\ & \\ & \end{array}$$