Finding derivative of x^x using limit definition

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To find the derivative of x^x using the limit definition, the limit lim [(x+h)^h - x^h]/h as h approaches 0 is crucial. It simplifies to ln(x) by applying the properties of logarithms and the exponential function. The discussion emphasizes using the expansion of e^(ln(x)) and the Taylor series for log(1+h) to derive the limit. Understanding these expansions helps clarify why the limit equals ln(x). This approach is essential for correctly calculating the derivative of x^x.
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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
 

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

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