# Derivative of x to the x

## Theorem

Let $x \in \R$ be a strictly positive real number.

Then:

$\dfrac {\mathrm d} {\mathrm d x} x^x = x^x \left({\ln x + 1}\right)$

## Proof 1

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

Define $y$ as $x^x$.

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 $\displaystyle \frac {\mathrm d} {\mathrm d x} \left({\ln y}\right)$ $=$ $\displaystyle \frac {\mathrm d} {\mathrm d x} \left({x \ln x}\right)$ Differentiating both sides with respect to $x$ $\displaystyle \frac 1 y \frac {\mathrm d y} {\mathrm d x}$ $=$ $\displaystyle \frac {\mathrm d} {\mathrm d x} \left({x \ln x}\right)$ Chain Rule and Derivative of Natural Logarithm Function $\displaystyle$ $=$ $\displaystyle \frac {\mathrm d} {\mathrm d x} \left({x}\right) \cdot \ln x + x \frac {\mathrm d} {\mathrm d x} \ln x$ 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 $\displaystyle \frac {\mathrm dy} {\mathrm dx}$ $=$ $\displaystyle x^x \left({\ln x + 1}\right)$ Multiply both sides by $y = x^x$

$\blacksquare$

## Proof 2

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

 $\displaystyle \frac {\mathrm d} {\mathrm d x} x^x$ $=$ $\displaystyle \frac {\mathrm d} {\mathrm d x} \exp \left({x \ln x}\right)$ Definition of Power $\displaystyle$ $=$ $\displaystyle \left({\frac {\mathrm d} {\mathrm d \left({x \ln x}\right)} \exp \left({x \ln x}\right)}\right) \left({\frac {\mathrm d} {\mathrm d x} \left({x \ln x}\right)}\right)$ Chain Rule $\displaystyle$ $=$ $\displaystyle \exp \left({x \ln x}\right) \left({\frac {\mathrm d} {\mathrm d x} \left({x \ln x}\right)}\right)$ Derivative of Exponential Function $\displaystyle$ $=$ $\displaystyle \exp \left({x \ln x}\right) \left({\frac {\mathrm d} {\mathrm d x} \left({x}\right) \cdot \ln x + x \frac {\mathrm d} {\mathrm d x} \ln x}\right)$ Product Rule for Derivatives $\displaystyle$ $=$ $\displaystyle \exp \left({x \ln x}\right) \left({1 \cdot \ln x + x \cdot \frac 1 x}\right)$ Derivative of Identity Function and Derivative of Natural Logarithm Function $\displaystyle$ $=$ $\displaystyle \exp \left({x \ln x}\right) \left({\ln x + 1}\right)$ Real Multiplication Identity is One and Inverses for Real Multiplication $\displaystyle$ $=$ $\displaystyle x^x \left({\ln x + 1}\right)$ Definition of Power

$\blacksquare$