Derivative of x to the x
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Theorem
Let $x \in \R$ be a real variable whose domain is the set of (strictly) positive real numbers $\R_{>0}$.
Then:
- $\dfrac \d {\d x} x^x = x^x \paren {\ln x + 1}$
Proof 1
Note that the Power Rule cannot be used because the index is not a constant.
Let $y := x^x$.
As $x$ was stipulated to be positive, we can take the natural logarithm of both sides:
\(\ds \ln y\) | \(=\) | \(\ds \ln x^x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x \ln x\) | Laws of Logarithms | |||||||||||
\(\ds \map {\frac \d {\d x} } {\ln y}\) | \(=\) | \(\ds \map {\frac \d {\d x} } {x \ln x}\) | differentiating both sides with respect to $x$ | |||||||||||
\(\ds \frac 1 y \frac {\d y} {\d x}\) | \(=\) | \(\ds \map {\frac \d {\d x} } {x \ln x}\) | Chain Rule for Derivatives and Derivative of Natural Logarithm Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\frac \d {\d x} } x \cdot \ln x + x \frac {\d} {\d x} \ln x\) | Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 \cdot \ln x + x \cdot \frac 1 x\) | Derivative of Identity Function and Derivative of Natural Logarithm Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln x + 1\) | simplification | |||||||||||
\(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds x^x \paren {\ln x + 1}\) | multiplying both sides by $y = x^x$ |
$\blacksquare$
Proof 2
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) |
$\blacksquare$
Proof 3
From Derivative of $x^{a x}$ we have:
- $\dfrac \d {\d x} x^{a x} = a x^{a x} \paren {\ln x + 1}$
The result follows on setting $a = 1$.
$\blacksquare$