设f(x)=(x-1)(x-2)...(x-2016),则f'(2016)=
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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 ′(2016) = = [(2016-1)(2016-2)...(2016-2016)] + [2016(2016-2)...(2016-2016)] + [2016(2016-1)...(2016-2016)] +......+ [2016(2016-1)...(2016-2015)]
= 0+0+......+0+2016!
= 2016!
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 ′(2016) = = [(2016-1)(2016-2)...(2016-2016)] + [2016(2016-2)...(2016-2016)] + [2016(2016-1)...(2016-2016)] +......+ [2016(2016-1)...(2016-2015)]
= 0+0+......+0+2016!
= 2016!
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