f(x)=x(x-1)(x-2)(x-3)(x-4)(x-5)的导数怎么求
答案:2 悬赏:50
解决时间 2021-01-24 15:27
- 提问者网友：沉默的哀伤
- 2021-01-24 12:21
f(x)=x(x-1)(x-2)(x-3)(x-4)(x-5)的导数怎么求
最佳答案
- 二级知识专家网友：污到你湿
- 2021-01-24 13:19
令 f(x)=y=x(x-1)(x-2)(x-3)(x-4)(x-5)
两边取对数得：
lny = lnx + ln(x-1) + ln(x-2) + ln(x-3) + ln(x-4)+ ln(x-5)
两边求导得：
1/y * y′ = 1/x + 1/(x-1) + 1/(x-2) + 1/(x-3) + 1/(x-4) + 1/(x-5)
f ′(x) = y′ = y { 1/x + 1/(x-1) + 1/(x-2) + 1/(x-3) + 1/(x-4) + 1/(x-5) }
= x(x-1)(x-2)(x-3)(x-4)(x-5) * { 1/x + 1/(x-1) + 1/(x-2) + 1/(x-3) + 1/(x-4) + 1/(x-5) }
= (x-1)(x-2)(x-3)(x-4)(x-5) + x(x-2)(x-3)(x-4)(x-5) + x(x-1)(x-3)(x-4)(x-5) + x(x-1)(x-2)(x-4)(x-5) + x(x-1)(x-2)(x-3)(x-5) + x(x-1)(x-2)(x-3)(x-4)
两边取对数得：
lny = lnx + ln(x-1) + ln(x-2) + ln(x-3) + ln(x-4)+ ln(x-5)
两边求导得：
1/y * y′ = 1/x + 1/(x-1) + 1/(x-2) + 1/(x-3) + 1/(x-4) + 1/(x-5)
f ′(x) = y′ = y { 1/x + 1/(x-1) + 1/(x-2) + 1/(x-3) + 1/(x-4) + 1/(x-5) }
= x(x-1)(x-2)(x-3)(x-4)(x-5) * { 1/x + 1/(x-1) + 1/(x-2) + 1/(x-3) + 1/(x-4) + 1/(x-5) }
= (x-1)(x-2)(x-3)(x-4)(x-5) + x(x-2)(x-3)(x-4)(x-5) + x(x-1)(x-3)(x-4)(x-5) + x(x-1)(x-2)(x-4)(x-5) + x(x-1)(x-2)(x-3)(x-5) + x(x-1)(x-2)(x-3)(x-4)
全部回答
- 1楼网友：街头电车
- 2021-01-24 14:50
f(x)=x(x-1)(x-2)(x-3)(x-4)(x-5)
lnf(x)=lnx+ln(x-1)+ln(x-2)+ln(x-3)+ln(x-4)+ln(x-5)
f'(x)/f(x)=1/x+1/(x-1)+1/(x-2)+1/(x-3)+1/(x-4)+1/(x-5)
∴f'(x)=f(x)[1/x+1/(x-1)+1/(x-2)+1/(x-3)+1/(x-4)+1/(x-5)]
=x(x-1)(x-2)(x-3)(x-4)(x-5)[1/x+1/(x-1)+1/(x-2)+1/(x-3)+1/(x-4)+1/(x-5)]
lnf(x)=lnx+ln(x-1)+ln(x-2)+ln(x-3)+ln(x-4)+ln(x-5)
f'(x)/f(x)=1/x+1/(x-1)+1/(x-2)+1/(x-3)+1/(x-4)+1/(x-5)
∴f'(x)=f(x)[1/x+1/(x-1)+1/(x-2)+1/(x-3)+1/(x-4)+1/(x-5)]
=x(x-1)(x-2)(x-3)(x-4)(x-5)[1/x+1/(x-1)+1/(x-2)+1/(x-3)+1/(x-4)+1/(x-5)]
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