Commit e5a34f
2024-10-19 13:36:07 Qwas: Save 数学公式| /dev/null .. "b/\351\253\230\346\225\260/\346\225\260\345\255\246\345\205\254\345\274\217.md" | |
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| + | # 数学公式 |
| + | |
| + | ## 泰勒公式 |
| + | |
| + | ```math |
| + | e^x = 1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!} |
| + | sinx = x-\frac{x^3}{3!}+...+(-1)^n\frac{x^{2n+1}}{(2n+1)!} |
| + | cosx = 1-\frac{x^2}{2!}+...+(-1)^n\frac{x^{2n}}{(2n)!} |
| + | ln(1+x) = x-\frac{x^2}{2}+...+(-1)^{n-1}\frac{x^n}{n} |
| + | \frac{1}{1-x} = 1+x+x^2+...+x^n$ ，|x|<1 |
| + | \frac{1}{1+x} = 1-x+x^2-...+(-1)^nx^n |
| + | (1+x)^a = 1+ax+\frac{a(a-1)}{2}x^2+O(x^2) |
| + | tanx = x+\frac{1}{3}x^3+O(x^3) |
| + | arcsinx = x+\frac{1}{6}x^3+O(x^3) |
| + | arctanx = x-\frac{1}{3}x^3+O(x^3) |
| + | ``` |
| + | |
| + | ## 高阶导数 |
| + | |
| + | ```math |
| + | a^{x^{(n)}} = a^x(lna)^n ，a>0, a\neq 1 |
| + | e^{x^{(n)}} = e^x |
| + | (sinkx)^{(n)} = k^nsin(kx+n\cdot \frac{\pi}{2}) |
| + | (coskx)^{(n)} = k^ncos(kx+n\cdot \frac{\pi}{2}) |
| + | (lnx)^{(n)} = (-1)^{n-1} \cdot \frac{(n-1)!}{x^n} |
| + | (\frac{1}{x})^{(n)} = (-1)^n \cdot \frac{n!}{x^{n+1}} |
| + | [ln(1+x)]^{(n)} = (-1)^{n-1} \cdot \frac{(n-1)!}{(1+x)^n} |
| + | (\frac{1}{1+a})^{(n)} = (-1)^n \cdot \frac{n!}{(x+a)^{n+1}} |
| + | [(x+x_0)^m]^{(n)} = m(m-1) \cdot \cdot \cdot (m-n+1)(x+x_0)^{m-n} |
| + | ``` |
| + | |
| + | ## 源码 |
| + | |
| + | ```txt |
| + | ## 泰勒公式 |
| + | e^x = 1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!} |
| + | sinx = x-\frac{x^3}{3!}+...+(-1)^n\frac{x^{2n+1}}{(2n+1)!} |
| + | cosx = 1-\frac{x^2}{2!}+...+(-1)^n\frac{x^{2n}}{(2n)!} |
| + | ln(1+x) = x-\frac{x^2}{2}+...+(-1)^{n-1}\frac{x^n}{n} |
| + | \frac{1}{1-x} = 1+x+x^2+...+x^n$ ，|x|<1 |
| + | \frac{1}{1+x} = 1-x+x^2-...+(-1)^nx^n |
| + | (1+x)^a = 1+ax+\frac{a(a-1)}{2}x^2+O(x^2) |
| + | tanx = x+\frac{1}{3}x^3+O(x^3) |
| + | arcsinx = x+\frac{1}{6}x^3+O(x^3) |
| + | arctanx = x-\frac{1}{3}x^3+O(x^3) |
| + | |
| + | ## 高阶导数 |
| + | a^{x^{(n)}} = a^x(lna)^n ，a>0, a\neq 1 |
| + | e^{x^{(n)}} = e^x |
| + | (sinkx)^{(n)} = k^nsin(kx+n\cdot \frac{\pi}{2}) |
| + | (coskx)^{(n)} = k^ncos(kx+n\cdot \frac{\pi}{2}) |
| + | (lnx)^{(n)} = (-1)^{n-1} \cdot \frac{(n-1)!}{x^n} |
| + | (\frac{1}{x})^{(n)} = (-1)^n \cdot \frac{n!}{x^{n+1}} |
| + | [ln(1+x)]^{(n)} = (-1)^{n-1} \cdot \frac{(n-1)!}{(1+x)^n} |
| + | (\frac{1}{1+a})^{(n)} = (-1)^n \cdot \frac{n!}{(x+a)^{n+1}} |
| + | [(x+x_0)^m]^{(n)} = m(m-1) \cdot \cdot \cdot (m-n+1)(x+x_0)^{m-n} |
| + | ``` |