Derivative & integral table

Detivative table

$$\begin{aligned} y = c \ \ \ \ &\ \ \ \ y' = 0\\ y = a^x \ \ \ \ &\ \ \ \ y' = a^x \ln a\\ y = \log_a x \ \ \ \ &\ \ \ \ y' = \frac{1}{x\ln a}\\ y = \ln x \ \ \ \ &\ \ \ \ y' = \frac{1}{x}\\ y = x^a \ \ \ \ &\ \ \ \ y' = ax^{a-1}\\ y = \sin x \ \ \ \ &\ \ \ \ y' = \cos x\\ y = \cos x \ \ \ \ &\ \ \ \ y' = -\sin x\\ y = \tan x \ \ \ \ &\ \ \ \ y' = \sec^2 x\\ y = \cot x \ \ \ \ &\ \ \ \ y' = -\csc^2 x\\ y = \arcsin x \ \ \ \ &\ \ \ \ y' = \frac{1}{\sqrt{1-x^2}}\\ y = \arccos x \ \ \ \ &\ \ \ \ y' = -\frac{1}{\sqrt{1-x^2}}\\ y = \arctan x \ \ \ \ &\ \ \ \ y' = \frac{1}{1+x^2}\\ y = \mathrm{arccot} x \ \ \ \ &\ \ \ \ y' = -\frac{1}{1+x^2} \end{aligned}$$

积分表

$$\begin{aligned} & \int k\mathrm{d}x = kx\\ & \int x^a \mathrm{d}x = \frac{1}{a+1} x^{a+1} &(a\neq -1) \ \ \\ & \int \frac{1}{x} \mathrm{d}x = \ln \lvert x\rvert \\ & \int a^x \mathrm{d}x = \frac{a^x}{\ln a} &(0 < a \neq 1)\\ & \int e^x \mathrm{d}x = e^x\\ & \int \sin x \mathrm{d}x = -\cos x \\ & \int \cos x \mathrm{d}x = \sin x\\ & \int \sec^2x \mathrm{d}x = \tan x\\ & \int \csc^2 x\mathrm{d}x = -\cot x \\ & \int \frac{\mathrm{d}x}{a^2 + x^2} = \frac{1}{a} \arctan \frac{x}{a} & (a\neq 0)\\ & \int \frac{\mathrm{d}x}{x^2 - a^2} = \frac{1}{2a}\ln \left\lvert \frac{x-a}{x+a} \right\rvert & (a\neq 0)\\ & \int \frac{\mathrm{d}x}{\sqrt{a^2 - x^2}} = \arcsin \frac{x}{a} & (a>0) \\ & \int \frac{\mathrm{d}x}{\sqrt{x^2 \pm a^2 }} = \ln \left\lvert x \pm \sqrt{x^2 \pm a^2} \right\rvert \end{aligned}$$

Snippets

$$\int{e^{ax}}\mathrm{d}x = \frac{1}{a} e^{ax}$$

Taylor expansion

$$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^3(a)}{3!}(x-a)^3 + \cdots + \frac{f^n(a)}{n!}(x-a)^n + \cdots$$

For example,

$$e^{x} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

$$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$

$$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$

积化和差 和差化积

$$\begin{aligned} sin(\alpha) cos(\beta) &= \frac{ sin(\alpha + \beta) + sin(\alpha - \beta) }{2}\\ cos(\alpha) sin(\beta) &= \frac{sin(\alpha + \beta) - sin(\alpha - \beta)}{2}\\ cos(\alpha) cos(\beta) &= \frac{cos(\alpha + \beta) + cos(\alpha - \beta)}{2}\\ sin(\alpha) sin(\beta) &= - \frac{cos(\alpha + \beta) - cos(\alpha - \beta)}{2} \end{aligned}$$

$$\begin{aligned} sin(\alpha) + sin(\beta) &= 2 sin(\frac{\alpha + \beta}{2}) cos(\frac{\alpha - \beta}{2})\\ sin(\alpha) - sin(\beta) &= 2 cos(\frac{\alpha + \beta}{2}) sin (\frac{\alpha - \beta}{2})\\ cos(\alpha) + cos(\beta) &= 2 cos(\frac{\alpha + \beta}{2}) cos (\frac{\alpha - \beta}{2})\\ cos(\alpha) - cos(\beta) &= -2sin(\frac{\alpha + \beta}{2}) sin (\frac{\alpha - \beta}{2}) \end{aligned}$$