Differentiate x to the power x^x.

Prologue

Let us begin with the well-known limit:

We can get easily from the first principle:

(1)

Differentiation Fun

Let y = x^x.

Then ln y = x ln x, and by (1):

(2)

Now let

ln y = x^x ln x, and by (2) we get

Hence,

Exercise

Verify: