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设函数f(x)=x(x-1)(x-2)...(x-2016),求f(0)的导数

2025-03-01 12:37:17

推荐回答（1个）

回答1：

令f(x)=x(x-1)(x-2)...(x-2016)=y
lny = lnx+ln(x-1)+ln(x-2)+...+ln(x-2016)
1/y * y′ = 1/x + 1/(x-1) + 1/(x-2) +......1/(x-2016)
y ′ = y * { 1/x + 1/(x-1) + 1/(x-2) +......1/(x-2016) }
= x(x-1)(x-2)...(x-2016) * { 1/x + 1/(x-1) + 1/(x-2) +......1/(x-2016) }
= [(x-1)(x-2)...(x-2016)] + [x(x-2)...(x-2016)] + [x(x-1)...(x-2016)] +......+ [x(x-1)...(x-2015)]

f ′(0) = = [(0-1)(0-2)...(0-2016)] + [0(0-2)...(0-2016)] + [0(0-1)...(0-2016)] +......+ [0(0-1)...(0-2015)]
= 2016! +0+0+......+0
= 2016!