Yes, you can use the same derailleur. The derailleur generally doesn't care how many cogs are on your freewheel, especially if you're using friction shifting. If you're using indexed shifting, you need to make sure the shifter matches the number of cogs and that the derailleur matches the cable pull ratio of the shifter.

Now while the derailleur doesn't care about the number of cogs, it does care about the size of the cogs. There are two specs the derailleur needs to meet: First, the derailleur needs to be able to handle the size of the freewheel's biggest cog. Secondly, it needs enough "wrap capacity" to handle chain slack. The wider your gear range, the more slack there is to handle in your lowest gear, and the more capacity the derailleur needs.