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(1)f(x)=x^u
证法一:(u为自然数)
f'(x)
=lim [(x+Δx)^u-x^u]/Δx
=lim (x+Δx-x)[(x+Δx)^(u-1)+x*(x+Δx)^(u-2)+...+x^(u-2)*(x+Δx)+x^(u-1)]/Δx
=lim [(x+Δx)^(u-1)+x*(x+Δx)^(u-2)+...+x^(u-2)*(x+Δx)+x^(u-1)]
=x^(u-1)+x*x^(u-2)+x^2*x^(u-3)+ ...x^(u-2)*x+x^(u-1)
=ux^(u-1)
证法二:(u为任意实数)
f(x)=x^u
luf(x)=ulux
(luf(x))'=(ulux)'
f'(x)/f(x)=u/x
f'(x)=u/x*f(x)
f'(x)=u/x*x^u
f'(x)=ux^(u-1)
1年前
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