  We consider a family of sup-functionals of (drifted) fractional Brownian
motion with Hurst parameter $H\in(0,1)$. This family includes, but is not
limited to: expected value of the supremum, expected workload, Wills
functional, and Piterbarg-Pickands constant. Explicit formulas for the
derivatives of these functionals as functions of Hurst parameter evaluated at
$H=\tfrac{1}{2}$ are established. In order to derive these formulas, we develop
the concept of derivatives of fractional $\alpha$-stable fields introduced by
Stoev \& Taqqu (2004) and propose Paley-Wiener-Zygmund representation of
fractional Brownian motion.
