If you know what the individual errors are, you can add them up without doing derivatives in most cases. Unfortunately, it involves some math that I can't recall at the moment. There's different procedures for additive, divisive, etc. errors, so it depends on how you're manipulating the data. Look in a statistics book or just search the web for 'additive errors' or 'adding errors'. It's in my analytical chem book, but that thing is 2100 miles away from me right now (a worthy place for it).