Question

d(x^x)/dx

Answer 1

And why? Also, what is d(x^x^x)/dx, if it will not be immediately obvious from the main question's answer.

Answer 2

writing y = x^x, we have
ln y = x . ln x

Now differentiate both sides wrt x

Answer 3

Surely that just gives us d(lny)/dx?

Answer 4

ln y = x lnx
differentiate both sides
dy/dx * 1/y = x * 1/x + ln x
dy/dx = ( 1 + lnx ) / 1/y
= y(1 + lnx)
= x^x ( 1 + ln x)

Answer 5

i would write
\[x^x=e^{x\ln(x)}\] and use the chain and product rule.
you will get the same thing, since all the work is in the product rule