Derivative of x to the x/Proof 2
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Theorem
- $\dfrac \d {\d x} x^x = x^x \paren {\ln x + 1}$
Proof
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) |
$\blacksquare$