Question

设函数f(x)=x*(x-1)*(x-2)...(x-99)(x-100),则f′(0)=

需要步骤 详细易懂的 谢了

Answer 1

解：因为f(x)=x*(x-1)*(x-2)...(x-99)(x-100)
所以f(0)=0
所以f′(0)=lim(x→0)[f(x)-f(0)]/(x-0)
=lim(x→0)f(x)/x
=lim(x→0)[x*(x-1)*(x-2)...(x-99)(x-100)]/x
=lim(x→0)(x-1)*(x-2)...(x-99)(x-100)
=(0-1)*(0-2)...(0-99)(0-100)
=100!

注n!=1*2*3*...*n（！是阶乘的意思）

Answer 2

设（x-1)*(x-2)...(x-99)(x-100)为y,则有乘法求导公式有，f'(x)=y+x*y',所以f'(0)=y(0)=100!.
同理可求f'(1),f'(2),f'(3),......