y的n阶导数记作

需要保证在x点n阶可导

几个无穷阶导数

1. $[e^x{]}^{(n)}=e^x$
2. $[sin(x){]}^{(n)}=sin(x+n\cdot \frac{\pi }{2})$
3. $[cos(x){]}^{(n)}=cos(x+n\cdot \frac{\pi }{2})$
4. $[ln(1+x){]}^{(n)}=(-1{)}^{n-1}\cdot \frac{(n-1)!}{(1+x{)}^n}$ (当 $n=1$ 时也成立)
5. $[x^{\mu }{]}^{(n)}=\mu (\mu -1)(\mu -2)...(\mu -n+1)\cdot x^{\mu -n}$
6. 满足可加性: $(u\pm v{)}^{(n)}=u^{(n)}\pm v^{(n)}$
7. 莱布尼兹公式$$ (u\cdot v)^{(n)}=\displaystyle \sum^{n}_{k=0} C^k_n \cdot u^{(n-k)}\pm v^{(k)}$$
   $n=1$ 时: $(u\cdot v{)}^{(1)}=u^{\prime }v+u{v}^{\prime }$
   $n=2$ 时: $(u\cdot v{)}^{(2)}=u^{″}v+2u^{\prime }{v}^{\prime }+u{v}^{″}$
   $n=3$ 时: $(u\cdot v{)}^{(3)}=u^{‴}v+3u^{″}{v}^{\prime }+3u^{\prime }{v}^{″}+u{v}^{‴}$