In this entry, we will derive differential equations^{} satisfied by the function ${x}^{x}$.
We begin by computing its derivative^{}. To do this, we write ${x}^{x}={e}^{x\mathrm{log}x}$ and
apply the chain rule^{}:

$$\frac{d}{dx}{x}^{x}=\frac{d}{dx}{e}^{x\mathrm{log}x}={e}^{x\mathrm{log}x}(1+\mathrm{log}x)={x}^{x}(1+\mathrm{log}x)$$ 

Set $y={x}^{x}$. Then we have ${y}^{\prime}/y=1+\mathrm{log}x$. Taking
another derivative, we have

$$\frac{d}{dx}\left(\frac{{y}^{\prime}}{y}\right)=\frac{1}{x}\cdot $$ 

Applying the quotient rule^{} and simplifying, this becomes

$$y{y}^{\prime \prime}{({y}^{\prime})}^{2}{y}^{2}/x=0.$$ 

It is also possible to derive an equation in which $x$ does not appear.
We start by noting that, if $z=1/x$, then ${z}^{\prime}+{z}^{2}=0$. If, as
above, $y={x}^{x}$, we have $(d/dx)({y}^{\prime}/y)=z$. Combining equations,

$$\frac{{d}^{2}}{d{x}^{2}}\left(\frac{{y}^{\prime}}{y}\right)+{\left(\frac{d}{dx}\left(\frac{{y}^{\prime}}{y}\right)\right)}^{2}=0;$$ 

applying the quotient rule and simplifying,

$${y}^{3}{y}^{\prime \prime \prime}{y}^{2}{({y}^{\prime \prime})}^{2}+2y{({y}^{\prime})}^{2}{y}^{\prime \prime}3{y}^{2}{y}^{\prime}{y}^{\prime \prime}{({y}^{\prime})}^{4}+2y{({y}^{\prime})}^{3}=0.$$ 
