基本运算

求导公式

基本求导

$$\begin{array}{lcl}{C}^{\prime }& & =0\\ (x^n{)}^{\prime }& & =n{x}^{n-1}\\ (sinx{)}^{\prime }& & =cosx\\ (cosx{)}^{\prime }& & =-sinx\\ (tanx{)}^{\prime }& & =se{c}^{2}x\\ (cotx{)}^{\prime }& & =-cs{c}^{2}x\\ (arcsinx{)}^{\prime }& & =\frac{1}{\sqrt{1-x^2}}\\ (arccosx{)}^{\prime }& & =-\frac{1}{\sqrt{1-x^2}}\\ (arctanx{)}^{\prime }& & =\frac{1}{1+x^2}\\ (arccotx{)}^{\prime }& & =-\frac{1}{1+x^2}\\ (e^x{)}^{\prime }& & =e^x\\ (a^x{)}^{\prime }& & =a^{x}lna\\ (lnx{)}^{\prime }& & =\frac{1}{x}\\ (lo{g}_{a}x{)}^{\prime }& & =\frac{1}{xlna}\end{array}$$

四则运算

$$\begin{array}{lcl}(u(x)\pm v(x){)}^{\prime }& & =u^{\prime }(x)\pm v^{\prime }(x)\\ (u(x)v(x){)}^{\prime }& & =u^{\prime }(x)v(x)+u(x)v^{\prime }(x)\\ (\frac{u(x)}{v(x)}{)}^{\prime }& & =\frac{u^{\prime }(x)v(x)-u(x)v^{\prime }(x)}{v^2(x)}\end{array}$$

反函数

$$f^{\prime }(x)=\underset{\Delta x\to 0}{\lim }\frac{\Delta y}{\Delta x}=\underset{\Delta x\to 0}{\lim }\frac{1}{\frac{\Delta x}{\Delta y}}=\frac{1}{{\lim }_{\Delta y\to 0}}\frac{\Delta x}{\Delta y}=\frac{1}{{\phi }^{\prime }(x)}$$

例：

$$y^{\prime }=(arcsinx{)}^{\prime }=\frac{1}{(\sin y{)}^{\prime }}=\frac{1}{\cos y}=\frac{1}{\sqrt{1-{\sin }^{2}y}}=\frac{1}{\sqrt{1-x^2}}$$

复合函数

$$\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}$$

幂指函数

$$\begin{array}{l}y=u(x{)}^{v(x)}\\ y^{\prime }=u(x{)}^{v(x)}(v^{\prime }(x)\ln u(x)+\frac{v(x)}{u(x)}{u}^{\prime }(x))\end{array}$$

隐函数

$$\begin{array}{l}\sin xy-\ln (x+y)=0\\ \cos xy\times (1\times y+x{y}^{\prime })-\frac{1}{x+y}(1+y^{\prime })=0\end{array}$$

参数式方程

$$\{\begin{array}{l}x=e^t\cos t,\\ y=e^t\sin t\end{array}$$ $$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

微积分

$$\begin{array}{lcl}\int kdx& & =kx+C\\ \int x^{\mu }dx& & =\frac{x^{\mu +1}}{\mu +1}+C,(\mu \ne -1)\\ \int \frac{1}{x}dx& & =\ln x+C\\ \int \frac{1}{1+x^2}dx& & =\arctan x+C\\ \int \frac{1}{\sqrt{1-x^2}}dx& & =\arcsin x+C\end{array}$$

极限

运算法则

• 两个无穷小的和是无穷小，有限个无穷小和是无穷小
• 有界与无穷小的积是无穷小
• 常数与无穷小的积是无穷小
• 有限个无穷小的积是无穷小

$$\begin{gathered} \lim f(x)=A,\lim g(x)=B \\ \lim (f(x)\pm g(x))=\lim f(x)\pm \lim g(x)=A\pm B \\ \lim (f(x)\cdot g(x))=\lim f(x)\cdot \lim g(x)=A\cdot B \\ \lim \frac{f(x)}{g(x)}=\frac{\lim f(x)}{\lim g(x)}(B\ne 0) \end{gathered}$$ $$\begin{gathered} \lim (Cf(x))=C\lim f(xC \\ \lim (f(x){)}^n=(\lim f(x){)}^n \\ 若f(x)\ge g(x),或f(x)>f(x),则\lim f(x)\ge \lim g(x) \end{gathered}$$