Cute calculus puzzle

Let

$f(x) = \mathrm{arctan}(x) + \mathrm{arctan}(1/x)$.

What is $\frac{d}{dx}f(x)$?  What does it say about $f(x)$?  Try to calculate some values of $f(x)$ and explain.

Author: hcsoso

Ph.D. student in the Department of Computer Science, University of Illinois at Urbana-Champaign.