[x^(1/x)]求导是多少啊?
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[x^(1/x)]求导是多少啊?
[x^(1/x)]求导是多少啊?
[x^(1/x)]求导是多少啊?
x^(1/x) = e^[1/x*lnx]
[x^(1/x)]' = [e^[1/x*lnx]]'=e^[1/x*lnx]*[1/x^2-1/x^2*lnx ]
=x^(1/x)*(1-lnx)/x^2
y=x^(1/x)
ln y=(ln x)/x ,
两边求导:y'/y=(1-ln x)/x^2
于是y'=y*(1-ln x)/x^2=(1-ln x)*x^(1/x-2).
幂指函数求导
[x^(1/x)]=e^ln[x^(1/x)]=e^[(1/x)lnx]
求导=e^[(1/x)lnx]*(lnx/(-x^2)+1/x^2)=[x^(1/x)]*(1-lnx)/x^2