if you push the derailleur cage forward and let go, it should spring back with no resistance and with a bit of force.
if it springs back but a bit too slowly and with some resistance, that will likely be the cause of your loose chain.
it's usually just due to the spring being caked in crap.
best thing is to just clean out all the gunk with a brake cleaner or something which will take old grease out with it, give it a good clean up. this will restore the snappiness. then relube with spray grease or something suitable.
if thats not the problem, you may have too many links in the chain