数学之家
标题:
Leibniz公式
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作者:
castelu
时间:
2017-11-8 19:00
标题:
Leibniz公式
设$y=uv$,则
$$y'=u'v+uv',$$
$$y''=\left( u'v+uv' \right)'=u''v+2u'v'+uv'',$$
$$y'''=\left( u''v+2u'v'+uv'' \right)'=u'''v+3u''v'+3u'v''+uv''',$$
如此下去,不难看到,计算结果与二项式$\left( u+v \right)^n$展开式极为相似,用数学归纳法,可得
$$\left( uv \right)^{\left( n \right)}=u^{\left( n \right)}v^{\left( 0 \right)}+C_n^1u^{\left( n-1 \right)}v^{\left( 1 \right)}+C_n^2u^{\left( n-2 \right)}v^{\left( 2 \right)} + \cdots + C_n^ku^{\left( n-k \right)}v^{\left( k \right)} + \cdots + u^{\left( 0 \right)}v^{\left( n \right)}$$$$=\sum\limits_{k = 0}^nC_n^ku^{\left( n-k \right)}v^{\left( k \right)},$$
其中$u^{\left( 0 \right)}=u$,$v^{\left( 0 \right)}=v$。
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