Question

F(x)=x(x-1)(x-2)(x-3)………(x-100),求F(1)的导数.具体点，有步骤！麻烦你你们了！

Answer 1

F'(x)=x'*(x-1)(x-2)………(x-100)+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-100)+……+x(x-1)(x-2)………(x-99)
除了第二项，其他都有x-1，则x=1时都等于0
所以F'(1)=1*(1-2)*(1-3)*……*(1-100)
=-99！

Answer 2

两边求对数

lnF(x)=ln(x(x-1)(x-2)(x-3)………(x-100))
=lnx+ln(x-1)+...+ln(x-100)

两边求导
F'(x)/F(x)=1/x+1/(x-1)+...+1/(x-100)

F'(x)=(1/x+1/(x-1)+...+1/(x-100))F(x)
=(x-1)(x-2)(x-3)...(x-100)+x(x-2)(x-3)...(x-100)+...

只有第二项不含x-1

所以是F'(1)=1(1-2)(1-3)...(1-100)=-99!