What's the derivative of x^e^x?

Put u = x^e^x

ln u = ln (x^e^x)

ln u = e^x ln x

d(ln u)/dx = d(e^x ln x)/dx

(1/u) (du/dx) = e^x [(d lnx)/dx] + ln x [(d e^x)/dx]

(1/x^e^x) (d x^e^x /dx) = e^x (1/x) + e^x ln x

(1/x^e^x) (d x^e^x /dx) = (e^x /x) (1 + x ln x)

d x^e^x /dx = (x^e^x e^x /x) (1 + x ln x)

d x^e^x /dx = x^(e^x - 1) e^x (1 + x ln x)