We report the discovery of rapid variations of a high-velocity CIV broad absorption line trough in the quasar SDSS J141007.74+541203.3. This object was intensively observed in 2014 as a part of the Sloan Digital Sky Survey Reverberation Mapping Project, during which 32 epochs of spectroscopy were obtained with the Baryon Oscillation Spectroscopic Survey spectrograph. We observe significant (>4sigma) variability in the equivalent width of the broad (~4000 km/s wide) CIV trough on rest-frame timescales as short as 1.20 days (~29 hours), the shortest broad absorption line variability timescale yet reported. The equivalent width varied by ~10% on these short timescales, and by about a factor of two over the duration of the campaign. We evaluate several potential causes of the variability, concluding that the most likely cause is a rapid response to changes in the incident ionizing continuum. If the outflow is at a radius where the recombination rate is higher than the ionization rate, the timescale of variability places a lower limit on the density of the absorbing gas of n_e > 3.9 x 10^5 cm^-3. The broad absorption line variability characteristics of this quasar are consistent with those observed in previous studies of quasars, indicating that such short-term variability may in fact be common and thus can be used to learn about outflow characteristics and contributions to quasar/host-galaxy feedback scenarios.
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