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e5a34f Qwas 2024-10-19 13:36:07 1 
# 数学公式
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## 泰勒公式
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```math
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e^x = 1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}
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sinx = x-\frac{x^3}{3!}+...+(-1)^n\frac{x^{2n+1}}{(2n+1)!}
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cosx = 1-\frac{x^2}{2!}+...+(-1)^n\frac{x^{2n}}{(2n)!}
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ln(1+x) = x-\frac{x^2}{2}+...+(-1)^{n-1}\frac{x^n}{n}
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\frac{1}{1-x} = 1+x+x^2+...+x^n$ ，|x|<1
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\frac{1}{1+x} = 1-x+x^2-...+(-1)^nx^n
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(1+x)^a = 1+ax+\frac{a(a-1)}{2}x^2+O(x^2)
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tanx = x+\frac{1}{3}x^3+O(x^3)
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arcsinx = x+\frac{1}{6}x^3+O(x^3)
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arctanx = x-\frac{1}{3}x^3+O(x^3)
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```
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## 高阶导数
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```math
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a^{x^{(n)}} = a^x(lna)^n ，a>0, a\neq 1
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e^{x^{(n)}} = e^x
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(sinkx)^{(n)} = k^nsin(kx+n\cdot \frac{\pi}{2})
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(coskx)^{(n)} = k^ncos(kx+n\cdot \frac{\pi}{2})
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(lnx)^{(n)} = (-1)^{n-1} \cdot \frac{(n-1)!}{x^n}
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(\frac{1}{x})^{(n)} = (-1)^n \cdot \frac{n!}{x^{n+1}}
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[ln(1+x)]^{(n)} = (-1)^{n-1} \cdot \frac{(n-1)!}{(1+x)^n}
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(\frac{1}{1+a})^{(n)} = (-1)^n \cdot \frac{n!}{(x+a)^{n+1}}
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[(x+x_0)^m]^{(n)} = m(m-1) \cdot \cdot \cdot (m-n+1)(x+x_0)^{m-n}
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```
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## 源码
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```txt
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## 泰勒公式
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e^x = 1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}
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sinx = x-\frac{x^3}{3!}+...+(-1)^n\frac{x^{2n+1}}{(2n+1)!}
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cosx = 1-\frac{x^2}{2!}+...+(-1)^n\frac{x^{2n}}{(2n)!}
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ln(1+x) = x-\frac{x^2}{2}+...+(-1)^{n-1}\frac{x^n}{n}
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\frac{1}{1-x} = 1+x+x^2+...+x^n$ ，|x|<1
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\frac{1}{1+x} = 1-x+x^2-...+(-1)^nx^n
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(1+x)^a = 1+ax+\frac{a(a-1)}{2}x^2+O(x^2)
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tanx = x+\frac{1}{3}x^3+O(x^3)
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arcsinx = x+\frac{1}{6}x^3+O(x^3)
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arctanx = x-\frac{1}{3}x^3+O(x^3)
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## 高阶导数
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a^{x^{(n)}} = a^x(lna)^n ，a>0, a\neq 1
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e^{x^{(n)}} = e^x
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(sinkx)^{(n)} = k^nsin(kx+n\cdot \frac{\pi}{2})
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(coskx)^{(n)} = k^ncos(kx+n\cdot \frac{\pi}{2})
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(lnx)^{(n)} = (-1)^{n-1} \cdot \frac{(n-1)!}{x^n}
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(\frac{1}{x})^{(n)} = (-1)^n \cdot \frac{n!}{x^{n+1}}
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[ln(1+x)]^{(n)} = (-1)^{n-1} \cdot \frac{(n-1)!}{(1+x)^n}
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(\frac{1}{1+a})^{(n)} = (-1)^n \cdot \frac{n!}{(x+a)^{n+1}}
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[(x+x_0)^m]^{(n)} = m(m-1) \cdot \cdot \cdot (m-n+1)(x+x_0)^{m-n}
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```