Question

关于导数的高数题目…求高人帮忙

f(x)=x(x-1)(x-2)…（x-100),求f'(0)

Answer 1

显然f(0)=0

下面令x→0，那么

f'(0) = lim (f(x)-f(0))/x
=lim x(x-1)...(x-100)/x
=lim (x-1)...(x-100) <这里约掉了x>
= (-1)*(-2)...(-100)
=100!

Answer 2

本题关键在于将 (x-1)(x-2)…（x-100) 视为一个函数 p(x)，
对 f(x) = x*p(x) 乘积求导,计算f'(0)，无需实际求出 p'(x).

f(x) = x * [(x-1)(x-2)…（x-100)]

f'(x) = (x)' * [(x-1)(x-2)…（x-100)] + x * [(x-1)(x-2)…（x-100)]'
= [(x-1)(x-2)…（x-100)] + x * [(x-1)(x-2)…（x-100)]'

f'(0) = [(-1)(-2)…（-100)] + 0 * [(x-1)(x-2)…（x-100)]'|x=0
= (-1)(-2)…（-100)
= 100!

Answer 3

f'(x)=x'*(x-1)(x-2)…（x-100)+x(x-1)'*(x-2)…（x-100)+……+x(x-1)(x-2)…（x-100)'
=(x-1)(x-2)…（x-100)+x(x-2)…（x-100)+……+x(x-1)(x-2)…（x-99)
后面每项都有x
则x=0时等于0
所以f'(0)=-1*(-2)*……(-100)=100！

Answer 4

f（x）=x^101+100次项+……+二次项+（-1）*（-2）*……（-100）x

f'（x）=x高次项+……+一次项+100！
f’（0）=100！