每天起床第一句，每天起床第一句 泰勒公式记一记 -。-
泰勒公式
ex=1+x+2!x2+...+n!xn
sinx=x−3!x3+...+(−1)n(2n+1)!x2n+1
cosx=1−2!x2+...+(−1)n(2n)!x2n
ln(1+x)=x−2x2+...+(−1)n−1nxn ，$-1< x\leqslant 1$
1−x1=1+x+x2+...+xn ，|x|<1
1+x1=1−x+x2−...+(−1)nxn
(1+x)a=1+ax+2a(a−1)x2+O(x2)
tanx=x+31x3+O(x3)
arcsinx=x+61x3+O(x3)
arctanx=x−31x3+O(x3)
高阶导数
ax(n)=ax(lna)n ，$a>0, a\neq 1$
ex(n)=ex
(sinkx)(n)=knsin(kx+n⋅2π)
(coskx)(n)=kncos(kx+n⋅2π)
(lnx)(n)=(−1)n−1⋅xn(n−1)!
(x1)(n)=(−1)n⋅xn+1n!
[ln(1+x)](n)=(−1)n−1⋅(1+x)n(n−1)!
(1+a1)(n)=(−1)n⋅(x+a)n+1n!
- * [(x+x0)m](n)=m(m−1)⋅⋅⋅(m−n+1)(x+x0)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}