What type of rear derailleur to buy? My cheap bike that I use to commute has been taking a lot of punishment (mostly from salt that's been used to melt the snow). I've been thinking about changing the rear derailleur (maybe not right now, but for the summer), but I'm not sure what replacement part to get. My bike has a 6 speed 14-28T cassette. 

This is a picture of my derailleur. I think it's a direct mount, so I was thinking about substituting it with the Shimano TX-35 Rear Derailleur Direct Mount.



And this is the replacement part that I'm thinking of buying.



Would the replacement part work, or do I need one with a hanger?

Thank you very much everyone.
The Tourney RD TX-35 should work fine. All Tourney level derailleurs work with 6 and 7 speeds (sprocket spacing is the same). The 35 without the extra hanger tab looks much less cheap in my opinion.

I agree with suggestion made in comments to replace the cassette and chain at the same time.