I know they are all the negative form of a verb, but I just don't know the difference. I still remember when I was learning Japanese, my teacher told me that ずに equals ないで in a more ancient way, but what about the other two? When should I use ないで, and when should I use なくて? And how about ず? Thank you.
なくて stands for a causal connection, that is, できなくて やめた means that you quit because you couldn't, while できないで やめた means that you quit it without accomplishing it.
ず can be interchangeable to either ずに（ないで） or なくて.