math derivative 7

Let u = x^x
ln u = x ln x
1/u(du/dx) = ln x + 1
so du/dx = u(ln x + 1).
y = x^x ln x
= u ln x
dy/dx = u/x + ln x ( du/dx)
= (x^x)/x + ln x[u(ln x + 1)]
= x^(x -1) + ln x(ln x + 1)x^x.