Some definitions:
Diligence: Diligence is steadfast application, assiduousness and industry; the virtue of hard work.
Due diligence: Due diligence is the necessary amount of diligence required in a professional activity to avoid being negligent.
Negligence: Negligence is a failure to exercise the care that a reasonably prudent person would exercise in like circumstances.
Gross negligence: Gross negligence is a legal concept which means serious carelessness. Negligence is the opposite of diligence, or being careful. The standard of ordinary negligence is what conduct one expects from the proverbial "reasonable person." By analogy, if somebody has been grossly negligent, that means they have fallen so far below the ordinary standard of care that one can expect, to warrant the label of being "gross."