Some interesting problems

Hi, I'm an engineering student in my 2nd year, not that that's relevant but nevertheless, I stumbled across some interesting problems today that may be a little thought provoking.

The first is finding the first derivative of x^x
My initial guess is using the original basic differentiation technique that is

d (x^n) / dx = n x^(n-1)

and using the chain rule I believe (although it is inconesquential in this case since dx/dx = 1)
I came up with the answer to d x^x / dx as

x(x^(x-1)) ... which just happens to be x^x unless my brain has gone wonkey, regardless it looks very wrong to me, so perhaps someone can set me straight on that ^-^

The other thing was regarding this, i^i (i denotes the imaginary unit)

A friend told me today that this has infinite solutions, and that they are all real. Is anyone aware of any simple proof of this?
Regard, Yax

y = x^x

1. Take natural log of both sides: ln(y) = ln(x^x) = x*ln(x)

2. Differentiate lhs: d/dx ln(y) = y'/y

3. Differentiate rhs using product rule: d/dx x*ln(x) = 1*ln(x) + x/x = ln(x) + 1

4. Set lhs = rhs: y'/y = ln(x) + 1

5. Rearrange: y' = y*(ln(x) + 1) = (x^x)*(ln(x) + 1)

Yes of course I see now. Taking logarithms makes sense. Now I feel like a plum. Any idea with the other thing? I find that quite intriuging, how such a grubby little thing like i^i has infinite real solutions.

Regarding iⁱ, see the Wikipedia article Imaginary unit
Particularly, the section on i and Euler's formula