泰勒公式

• $e^x = 1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}$
• $sinx = x-\frac{x^3}{3!}+...+(-1)^n\frac{x^{2n+1}}{(2n+1)!}$
• $cosx = 1-\frac{x^2}{2!}+...+(-1)^n\frac{x^{2n}}{(2n)!}$
• $ln(1+x) = x-\frac{x^2}{2}+...+(-1)^{n-1}\frac{x^n}{n}$ ，$-1< x\leqslant 1$
• $\frac{1}{1-x} = 1+x+x^2+...+x^n$ ，|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

• $a^{x^{(n)}} = a^x(lna)^n$ ，$a>0, a\neq 1$
• $e^{x^{(n)}} = e^x$
• $(sinkx)^{(n)} = k^nsin(kx+n\cdot \frac{\pi}{2})$
• $(coskx)^{(n)} = k^ncos(kx+n\cdot \frac{\pi}{2})$

• $(lnx)^{(n)} = (-1)^{n-1} \cdot \frac{(n-1)!}{x^n}$
• $(\frac{1}{x})^{(n)} = (-1)^n \cdot \frac{n!}{x^{n+1}}$
• $[ln(1+x)]^{(n)} = (-1)^{n-1} \cdot \frac{(n-1)!}{(1+x)^n}$
• $(\frac{1}{1+a})^{(n)} = (-1)^n \cdot \frac{n!}{(x+a)^{n+1}}$

• * $[(x+x_0)^m]^{(n)} = m(m-1) \cdot \cdot \cdot (m-n+1)(x+x_0)^{m-n}$

源码区(LaTeX)

// 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}