  We report the results of a one year Swift X-ray/UV/optical programme
monitoring the dwarf Seyfert nucleus in NGC 4395 in 2008-2009. The UV/optical
flux from the nucleus was found to vary dramatically over the monitoring
period, with a similar pattern of variation in each of the observed UV/optical
bands (spanning 1900 - 5500 {\AA}). In particular, the luminosity of NGC 4395
in the 1900 {\AA} band changed by more than a factor of eight over the
monitoring period. The fractional variability was smaller in the UV/optical
bands than that seen in the X-rays, with the X-ray/optical ratio increasing
with increasing flux. Pseudo-instantaneous flux measurements in the X-ray and
each UV/optical band were well correlated, with cross correlation coefficients
of >0.7, significant at 99.9 per cent confidence. Archival Swift observations
from 2006 sample the intra-day X-ray/optical variability on NGC 4395. These
archival data show a very strong correlation between the X-ray and b bands,
with a cross-correlation coefficient of 0.84 (significant at >99 per cent
confidence). The peak in the cross correlation function is marginally resolved
and asymmetric, suggesting that X-rays lead the b band, but by 1 hour. In
response to recent (August 2011) very high X-ray flux levels from NGC4395 we
triggered Swift ToO observations, which sample the intra-hour X-ray/UV
variability. These observations indicate, albeit with large uncertainties, a
lag of the 1900 {\AA} band behind the X-ray flux of ~400 s. The tight
correlation between the X-ray and UV/optical lightcurves, together with the
constraints we place on lag time-scale are consistent with the UV/optical
variability of NGC 4395 being primarily due to reprocessing of X-ray photons by
the accretion disc.
