  We study the multi-wavelength variability of the blazar Mrk 421 at minutes to
days timescales using simultaneous data at $\gamma$-rays from Fermi, 0.7-20 keV
energies from AstroSat, and optical and near-infrared (NIR) wavelengths from
ground-based observatories. We compute the shortest variability timescales at
all of the above wavebands and find its value to be ~1.1 ks at the hard X-ray
energies and increasingly longer at soft X-rays, optical and NIR wavelengths as
well as at the GeV energies. We estimate the value of the magnetic field to be
0.5 Gauss and the maximum Lorentz factor of the emitting electrons ~1.6 x
$10^5$ assuming that synchrotron radiation cooling drives the shortest
variability timescale. Blazars vary at a large range of timescales often from
minutes to years. These results, as obtained here from the very short end of
the range of variability timescales of blazars, are a confirmation of the
leptonic scenario and in particular the synchrotron origin of the X-ray
emission from Mrk 421 by relativistic electrons of Lorentz factor as high as
$10^5$. This particular mode of confirmation has been possible using minutes to
days timescale variability data obtained from AstroSat and simultaneous
multi-wavelength observations.
