Derivative of x to the x

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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

$\blacksquare$


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