  Let $(Z_t)_{t\geq 0}$ denote the derivative martingale of branching Brownian
motion, i.e.\@ the derivative with respect to the inverse temperature of the
normalized partition function at critical temperature. A well-known result by
Lalley and Sellke [\textit{Ann. Probab.}, 15(3):1052--1061, 1987] says that
this martingale converges almost surely to a limit $Z_\infty$, positive on the
event of survival. In this paper, our concern is the fluctuations of the
derivative martingale around its limit. A corollary of our results is the
following convergence, confirming and strengthening a conjecture by Mueller and
Munier [\textit{Phys. Rev. E}, 90:042143, 2014]: \[ \sqrt{t} \left( Z_\infty -
Z_t + \frac{\log t}{\sqrt{2\pi t}} Z_\infty \right) \xrightarrow[t\to\infty]{}
S_{Z_\infty}, \quad \text{in law}, \] where $S$ is a spectrally positive
1-stable L\'evy process independent of $Z_\infty$.
  In a first part of the paper, a relatively short proof of (a slightly
stronger form of) this convergence is given based on the functional equation
satisfied by the characteristic function of $Z_\infty$ together with tail
asymptotics of this random variable. We then set up more elaborate arguments
which yield a more thorough understanding of the trajectories of the particles
contributing to the fluctuations. In this way, we can upgrade our convergence
result to functional convergence. This approach also sets the ground for a
follow-up paper, where we study the fluctuations of more general functionals
including the renormalized critical additive martingale.
  All proofs in this paper are given under the hypothesis $E[L(\log L)^3] <
\infty$, where the random variable $L$ follows the offspring distribution of
the branching Brownian motion. We believe this hypothesis to be optimal.
