Why does $\frac{x^n}{n^x}$ stop growing at the approximate value of $\pi (n)$?

Question

I noticed while playing around with these functions that $n^x$ will start slow and then speed up really fast in its growth rate. While $x^n$ grows more slowly, but faster than $n^x$ at the start. Coincidentally, $\frac {d}{dx} \frac{x^n}{n^x}$ will equal zero at $\frac {n}{\ln n}~\pi (n)$
Is there a reason for this or is it just a coincidence?