Differentiating a Super Exponential

The function  
  us a super exponential and can be differentiated by writing  
\[f(x)=e^{ln(x^x}=e^{x lnx}\]
  and using a combination of The Chain Rule and Differentiation - The Product Rule.
Using the product rule to differentiate  
\[x \times \frac{1}{x} +1 \times lnx=1+lnx\]
Using the chain rule to differentiate  
\[e^{x lnx}\]

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