DX is the same as DC - it refers to the image circle projected by the lens onto the sensor and matches the smaller size crop sensors found in most DSLRs
from sigma for DC
DC (DC Lens)
These are special lenses designed so that the image circle matches the smaller size of the image sensor of most digital SLR cameras. Their specialized design gives these lenses the ideal properties for digital cameras, the compact and lightweight construction is an added bonus ! including compact and lightweight construction.
From Wiki for Nikon DX
The Nikon DX format is an image sensor format of approximately 24×16 mm. Its dimensions are about 2/3 those of the 35mm film format (29mm vs 43mm diagonal, approx.). The format was created by Nikon for its digital SLR cameras, many of which are equipped with DX-sized sensors. DX format is very similar in size to sensors referred to by other camera manufacturers (Canon, Pentax, and Sony) as APS-C, so the terms are sometimes used interchangeably.
Both will work on 35mm film - but both will cause vignetting. The image circle will be recorded if its projected onto film or sensor, but more modern lenses will be optimised for a digital sensor
Cheers
Hugh