Finding derivative of x^x using limit definition

[scottshannon]

I am trying to find the derivative of x^x using the limit definition and am unable to follow what I have read. Can someone help me understand why lim [(x+h)^h -1]/h as h ---> 0 = ln(x). This part of the derivatio

 

[phion]

Try using the fact that e^{ln(x)}=x

 

[Qwertywerty]

Limit (a∧h) = 1 + h×log(a)/1! + ((h×log(a))∧2)/2! + ...
h→0

For the steps performed on the right side , use , as already said , x = e∧log(x) ,
and

Limit (log(1+h)) = h - (h∧2)/2 + (h∧3)/3 - ...
h→0

*Log is taken to base e .