Understanding the graph for $x^y = y^x$ I graphed $x^y=y^x$ and it is a union of the line y=x, with some other curve.  So my first question is, how do I derive that other curve? My next question is, why don't I get the same graph when I plot $x^{1/x}=y^{1/y}$?

Let $y=\mathcal{R}x$, $\mathcal{R}$ a variable. Then,
$$x^{\mathcal{R}x}=(\mathcal{R}x)^x\\
\implies \mathcal{R}x\ln x=x\ln (\mathcal{R}x)=x\ln \mathcal{R}+x\ln x\\
\implies (\mathcal{R}-1)\ln x=\ln \mathcal{R}$$
We can choose $\mathcal{R}=1$, or, if $\mathcal{R}>1$,
$$x=\exp\left(\dfrac{\ln \mathcal{R}}{r-1}\right)=\mathcal{R}^{1/(\mathcal{R}-1)}$$
I.e.,
$$y=\mathcal{R}^{\mathcal{R}/(\mathcal{R}-1)}$$
The non-straight graph is the graph for $y$ for which $\mathcal{R}\neq 1$.
Since $\mathcal{R}x\neq \dfrac{1}{x}$, you don't get the same graph.
