Derogative vs Offensive

Derogative comments can be offensive, depending on context - but they are certainly not synonyms. There can be a lot of overlap between the two, trying to explain the differences with examples:

Derogative not offensive : Consider the example of 'housewife' given in MW definition for derogative - it might be 'belittling' or w/e (personally I disagree but it does have negative connotations for some people) but you can hardly say that calling someone a housewife is 'offensive'.

Offensive not derogative : Some terms such as cripple are recently considered offensive, with people preferring to substitute 'handicapped' or 'special needs' etc. as terms. However as a statement of fact it isn't derogative (you aren't belittling someone by pointing out they can't walk properly)

Both derogative & offensive : See the list of ethnic slurs on Wikipedia, most of them are both derogative (having some negative connotation) and offensive for various reasons to the target populace.

They are not synonyms. A comment can be offensive in many ways, eg by using explicit sexual or profane language in an inappropriate setting, without it being derogatory to any one particular person or thing.

Derogative in Merriam-Webster:

...intended to make a person or thing seem of little importance or value.

Derogatory seems to be more usual in BrE.