Derivative of x to the x/Proof 2

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$\dfrac \d {\d x} x^x = x^x \paren {\ln x + 1}$


Note that the Power Rule cannot be used because the index is not a constant.

\(\ds \frac \d {\d x} x^x\) \(=\) \(\ds \frac \d {\d x} \map \exp {x \ln x}\) Definition 1 of Power (Algebra)
\(\ds \) \(=\) \(\ds \paren {\frac \d {\map \d {x \ln x} } \map \exp {x \ln x} } \paren {\map {\frac \d {\d x} } {x \ln x} }\) Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \map \exp {x \ln x} \paren {\map {\frac \d {\d x} } {x \ln x} }\) Derivative of Exponential Function
\(\ds \) \(=\) \(\ds \map \exp {x \ln x} \paren {\map {\frac \d {\d x} } x \cdot \ln x + x \frac \d {\d x} \ln x}\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds \map \exp {x \ln x} \paren {1 \cdot \ln x + x \cdot \frac 1 x}\) Derivative of Identity Function and Derivative of Natural Logarithm Function
\(\ds \) \(=\) \(\ds \map \exp {x \ln x} \paren {\ln x + 1}\) Real Multiplication Identity is One and Inverse for Real Multiplication
\(\ds \) \(=\) \(\ds x^x \paren {\ln x + 1}\) Definition 1 of Power (Algebra)