# 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:

\(\displaystyle \ln y\) | \(=\) | \(\displaystyle \ln x^x\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x \ln x\) | Laws of Logarithms | ||||||||||

\(\displaystyle \map {\frac \d {\d x} } {\ln y}\) | \(=\) | \(\displaystyle \map {\frac \d {\d x} } {x \ln x}\) | differentiating both sides with respect to $x$ | ||||||||||

\(\displaystyle \frac 1 y \frac {\d y} {\d x}\) | \(=\) | \(\displaystyle \map {\frac \d {\d x} } {x \ln x}\) | Chain Rule for Derivatives and Derivative of Natural Logarithm Function | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map {\frac \d {\d x} } x \cdot \ln x + x \frac {\d} {\d x} \ln x\) | Product Rule for Derivatives | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1 \cdot \ln x + x \cdot \frac 1 x\) | Derivative of Identity Function and Derivative of Natural Logarithm Function | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \ln x + 1\) | simplification | ||||||||||

\(\displaystyle \frac {\d y} {\d x}\) | \(=\) | \(\displaystyle x^x \paren {\ln x + 1}\) | 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.

\(\displaystyle \frac \d {\d x} x^x\) | \(=\) | \(\displaystyle \frac \d {\d x} \map \exp {x \ln x}\) | Definition 1 of Power (Algebra) | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {\frac \d {\map \d {x \ln x} } \map \exp {x \ln x} } \paren {\map {\frac \d {\d x} } {x \ln x} }\) | Chain Rule for Derivatives | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \exp {x \ln x} \paren {\map {\frac \d {\d x} } {x \ln x} }\) | Derivative of Exponential Function | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \exp {x \ln x} \paren {\map {\frac \d {\d x} } x \cdot \ln x + x \frac \d {\d x} \ln x}\) | Product Rule for Derivatives | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \exp {x \ln x} \paren {1 \cdot \ln x + x \cdot \frac 1 x}\) | Derivative of Identity Function and Derivative of Natural Logarithm Function | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map \exp {x \ln x} \paren {\ln x + 1}\) | Real Multiplication Identity is One and Inverses for Real Multiplication | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x^x \paren {\ln x + 1}\) | 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 $a = 1$.

$\blacksquare$