They are very similar, but a tiny bit different. Here are some examples of how they are used. derive... from gain... by gain... from gain...something I have gained weight. I have derived. (INCORRECT) Do you know how to derive the quadratic formula? I stand to gain some money with the sale of the house. What can you possibly gain by visiting that rude woman? I hate competing; I have never derived any pleasure from seeing another person lose.

They can be synonyms, but only in a narrow sense. Usually their meaning or common useage is quite different. And when they are synonyms, usually "gain" or a form of that is the word you will hear and not "derive". For instance, an example of them being synonyms is "I derived no pleasure from that" = "I gained no pleasure from that" but is the 2nd version you will hear in conversation the vast majority of times. The common usage of "derive" is more in academic or formal settings. An example in mathematics: "The sum is derived from adding A plus B". Or another example in business, "He derived his profits from the sale of ...". In both these examples, though, you would tend to use "gain" when speaking. "He gained his profits from the sale of ...", etc.