F(x)=x(x-1)(x-2)(x-3)………(x-100),求F(1)的导数.具体点,麻烦你你们了!
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F(x)=x(x-1)(x-2)(x-3)………(x-100),求F(1)的导数.具体点,麻烦你你们了!
![F(x)=x(x-1)(x-2)(x-3)………(x-100),求F(1)的导数.具体点,麻烦你你们了!](/uploads/image/z/19498631-23-1.jpg?t=F%28x%29%3Dx%28x-1%29%28x-2%29%28x-3%29%E2%80%A6%E2%80%A6%E2%80%A6%28x-100%29%2C%E6%B1%82F%281%29%E7%9A%84%E5%AF%BC%E6%95%B0.%E5%85%B7%E4%BD%93%E7%82%B9%2C%E9%BA%BB%E7%83%A6%E4%BD%A0%E4%BD%A0%E4%BB%AC%E4%BA%86%21)
两边求对数
lnF(x)=ln(x(x-1)(x-2)(x-3)………(x-100))
=lnx+ln(x-1)+...+ln(x-100)
两边求导
F'(x)/F(x)=1/x+1/(x-1)+...+1/(x-100)
F'(x)=(1/x+1/(x-1)+...+1/(x-100))F(x)
=(x-1)(x-2)(x-3)...(x-100)+x(x-2)(x-3)...(x-100)+...
只有第二项不含x-1
所以是F'(1)=1(1-2)(1-3)...(1-100)=-99!
lnF(x)=ln(x(x-1)(x-2)(x-3)………(x-100))
=lnx+ln(x-1)+...+ln(x-100)
两边求导
F'(x)/F(x)=1/x+1/(x-1)+...+1/(x-100)
F'(x)=(1/x+1/(x-1)+...+1/(x-100))F(x)
=(x-1)(x-2)(x-3)...(x-100)+x(x-2)(x-3)...(x-100)+...
只有第二项不含x-1
所以是F'(1)=1(1-2)(1-3)...(1-100)=-99!
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