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.
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