Question

用导数的定义求函数f(x)=1/√x在x=1处的导数

Answer 1

f'(x)=lim[f(1)-f(x)]/(1-x) 当x趋于1时
=lim(1-1/√x)/(1-x)
=lim(-1/2√x)/(-1)
=1/2

Answer 2

f(1)=1/√1=1

f'(1)=lim(x->0) (f(x+1)-f(1))/x=lim(x->0) (1/√(x+1) - 1)/x=lim(x->0) (1-√(x+1))/x√(x+1)= lim(x->0) -1/2*x/x√(x+1)=-1/2lim(x->0) 1/√(x+1)=-1/2*1=-1/2

(用了等价无穷小1-√(x+1)=-1/2*x)

Answer 3

f'(1)=lim<x->1> [f(x)-f(1)]/(x-1)=lim<x->1> [[(1/√x)-1]/(x-1)=lim<x->1> -1/[(√x+1)*√x]=-1/2.