Chapter 7 - Exponential Functions - 7.3 Logarithms and Their Derivatives - Exercises - Page 343: 78

$$ f'(x) =e^{ x^{x }}(x^{x})(\ln x+1).$$

Recall that $(e^x)'=e^x $, $(\ln x)'=\dfrac{1}{x}$. We have $$ f(x)=e^{x^x}=e^{ e^{x\ln x}}.$$ Now taking the derivative, we get $$ f'(x)= e^{ e^{x\ln x}}(e^{x\ln x})'=e^{ e^{x\ln x}}(e^{x\ln x})(\ln x+1)\\ =e^{ x^{x }}(x^{x})(\ln x+1).$$