Fx=x(x-1)(x-2)(x-3)…(x-1000) 求 F'(0)

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Fx=x(x-1)(x-2)(x-3)…(x-1000)求F''(0)Fx=x(x-1)(x-2)(x-3)…(x-1000)求F''(0)Fx=x(x-1)(x-2)(x-3)…(x-1000)求F''(

Fx=x(x-1)(x-2)(x-3)…(x-1000) 求 F'(0)
Fx=x(x-1)(x-2)(x-3)…(x-1000) 求 F'(0)

Fx=x(x-1)(x-2)(x-3)…(x-1000) 求 F'(0)
F'(x)=x'*(x-1)(x-2)(x-3)…(x-1000)+x(x-1)'*(x-2)(x-3)…(x-1000)+…+x(x-1)(x-2)(x-3)…(x-1000)'
=(x-1)(x-2)(x-3)…(x-1000)+x(x-2)(x-3)…(x-1000)+…+x(x-1)(x-2)(x-3)…(x-999)
除了第一项,后面都有x,所以x=0时都等于0
所以F'(0)=(-1)(-2)(-3)…(-1000)=1000!

Fx=x(x-1)(x-2)(x-3)…(x-1000) =ax^1001+bx^1000+…+kx^2+hx
对Fx求导之后代入x=0,显然含有x的项都等于0,即 F'(0) =h=(-1)(-2)…(-1000)=1000!