常见函数导数

$$
\begin{aligned}
&\frac{d}{dx} (x^n) = nx^{n-1} \\
&\frac{d}{dx} (\ln x) = \frac{1}{x} \\
&\frac{d}{dx} (e^x) = e^x \\
&\frac{d}{dx} (a^x) = a^x \ln a \\
&\frac{d}{dx} (\sin x) = \cos x \\
&\frac{d}{dx} (\cos x) = -\sin x \\
&\frac{d}{dx} (\tan x) = \sec^2 x \\
&\frac{d}{dx} (\cot x) = -\csc^2 x \\
&\frac{d}{dx} (\sec x) = \sec x \tan x \\
&\frac{d}{dx} (\csc x) = -\csc x \cot x \\
&\frac{d}{dx} (\arcsin x) = \frac{1}{\sqrt{1-x^2}} \\
&\frac{d}{dx} (\arccos x) = -\frac{1}{\sqrt{1-x^2}} \\
&\frac{d}{dx} (\arctan x) = \frac{1}{1+x^2} \\
&\frac{d}{dx} (\text{arccot } x) = -\frac{1}{1+x^2} \\
&\frac{d}{dx} (\text{arcsec } x) = \frac{1}{|x|\sqrt{x^2-1}} \\
&\frac{d}{dx} (\text{arccsc } x) = -\frac{1}{|x|\sqrt{x^2-1}}\\
\end{aligned}
$$

常见函数积分

$$
\begin{aligned}
& \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1 \\
& \int \frac{1}{x} \, dx = \ln |x| + C \\
& \int e^x \, dx = e^x + C \\
& \int a^x \, dx = \frac{a^x}{\ln a} + C, \quad a > 0, a \neq 1 \\
& \int \sin x \, dx = -\cos x + C \\
& \int \cos x \, dx = \sin x + C \\
& \int \sec^2 x \, dx = \tan x + C \\
& \int \csc^2 x \, dx = -\cot x + C \\
& \int \sec x \tan x \, dx = \sec x + C \\
& \int \csc x \cot x \, dx = -\csc x + C \\
& \int \frac{1}{1+x^2} \, dx = \arctan x + C \\
& \int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin x + C
\end{aligned}
$$

泰勒展开

$$\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots \quad \text{for} \quad |x| < 1$$

$$(1+x)^a = \sum_{n=0}^{\infty} \binom{a}{n} x^n = 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-1)(a-2)}{3!}x^3 + \cdots$$

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

$$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$

$$\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$$

$$\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$$

三角函数相关公式

和差化积

$$
\begin{aligned}
& \sin A + \sin B = 2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right) \\
& \sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right) \\
& \cos A + \cos B = 2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right) \\
& \cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)
\end{aligned}
$$

积化和差

$$
\begin{aligned}
& \sin A \cos B = \frac{1}{2} [\sin (A+B) + \sin (A-B)] \\
& \cos A \sin B = \frac{1}{2} [\sin (A+B) - \sin (A-B)] \\
& \cos A \cos B = \frac{1}{2} [\cos (A+B) + \cos (A-B)] \\
& \sin A \sin B = \frac{1}{2} [\cos (A-B) - \cos (A+B)]
\end{aligned}
$$