Question

求函数f(x)=x(x-1)(x-2)(x-3)…(x-100)，当x=0的导数

Answer 1

f'(x)=x'(x-1)(x-2)(x-3)…(x-100)+x(x-1)'(x-2)(x-3)…(x-100)+…+x(x-1)(x-2)(x-3)…(x-100)'
=(x-1)(x-2)(x-3)…(x-100)+x(x-2)(x-3)…(x-100)+…+x(x-1)(x-2)(x-3)…(x-99)
后面每项都有x，所以x=0时都等于0
所以f'(0)=(-1)(-2)(-3)……(-100)=100 !

Answer 2

对f(x)求一阶导数就行了
只是把后面99个式子的乘积看做一项就行了
即f'(x)=(x-1)(x-2)(x-3)…(x-100)+x[(x-1)(x-2)(x-3)…(x-100)]'
令x=0,则上式=(0-1)(0-2)(0-3)…(0-100）=100！