求导F(X)=(X^x)^xF(X)=(X^x)^x详细点,注意符号使用,免得我看不懂

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)
F(x) = y^x
两边取自然对数
lnF(x) = x * lny
两边同时求导
F'(x)/F(x) = x' * lny + x*(lny)'
F'(x)/F(x) = lny + x * (y'/y)
将 y' = x^x (lnx + 1) 以及 y = x^x 代入上式
F'(x)/F(x) = lnx^x + x * x^x * (lnx + 1)/x^x
F'(x)/F(x) = x lnx + x(lnx + 1)
F'(x)/F(x) = 2x lnx + x
F'(x) = (2x lnx + x) * F(x) = (x^x)^x (2x lnx + x)

令大X=t
F(t) = (t^x)^x = t^(x^2)
dF/dt = x^2 * t^( x^2 - 1 )