对函数f(x)=x^x 求导

用对数法
y=x^x
=e^[ln(x^x)]
=e^(xlnx)

令u=xlnx,则y=e^u
y'=(x^u)'•u'
=(e^u)•(xlnx)'
=[e^(xlnx)]•[x'lnx+x(lnx)']
=[e^(xlnx)]•(lnx+x•1/x)
=(x^x)(1+lnx)

设 y = x^x
取自然对数
lny = lnx^x
lny = xlnx
两边同时求导
(lny)' = (x lnx)'
y'/y = x'*lnx + x(lnx)'
y'/y = lnx + x/x
y'/y = lnx + 1
y' = (lnx + 1)*y = x^x (lnx + 1)

两边都取Ln
lnf（x)=xlnx
两边都取导数得
f'(x)/f(x)=lnx+1
f'(x)=(lnx+1)f(x)=x^x(lnx+1)

f(x)’=x * x^x-1