求函数Y=(x-1)(x-2)……(x-100)的导数（X大于100）求函数y=(x-1)(x-2).(x-100)(x＞100)的导数两边取自然对数得lny=ln(x-1)+ln(x-2)+.+ln(x-100)两边对x取导y'/y=(1/x-1)+(1/x-2)+.+(1/x-100)可两边怎么对X求导?

Question

求函数Y=(x-1)(x-2)……(x-100)的导数（X大于100）
求函数y=(x-1)(x-2).(x-100)(x＞100)的导数
两边取自然对数得
lny=ln(x-1)+ln(x-2)+.+ln(x-100)
两边对x取导y'/y=(1/x-1)+(1/x-2)+.+(1/x-100)
可两边怎么对X求导?

Answer 1

已知lnx对x求导为1/x
lny=ln(x-1)+ln(x-2)+.+ln(x-100)
lny对x求导(lny)'先对中间变量y求导,y再对x求导
即为y'/y
ln(x-1)+ln(x-2)+.+ln(x-100)对x求导
和的导数等于导数的和
[ln(x-1)+ln(x-2)+.+ln(x-100)]'=[ln(x-1)]'+[ln(x-2)]'+...+[ln(x-100)]'
分别把x-1,x-2,...,x-100看成中间变量,
先对中间变量求导,中间变量再对x求导
[ln(x-1)]'+[ln(x-2)]'+...+[ln(x-100)]'=1/(x-1)+1/(x-2)+...+1/(x-100)
所以
y'/y=1/(x-1)+1/(x-2)+...+1/(x-100)
y'=y[1/(x-1)+1/(x-2)+...+1/(x-100)]
y'=(x-1)(x-2).(x-100)[1/(x-1)+1/(x-2)+...+1/(x-100)]