"Rogative" root (as in prerogative, derogative, interrogative)

Solution 1:

Yes: the root is ultmately rogare, "ask".

Interrogative: asking at intervals, or between people.

Prerogative: this comes from Latin, "to be asked first" and connotes privilege.

Derogative: this means partial abrogation. Abrogation comes directly from a Latin root abrogare "to repeal, to disregard, ignore, repudiate, to cancel, revoke, to take away" (OED). This is how [for example] the UK can implement a derogation from European Union law: it disregards the undesirable part of it.

The meaning of derogative in the sense of "belittling" is an extension of this, detracting from or disregarding reputation.

Solution 2:

Their roots are all in the Latin word, rogāre, to ask. They all have to do with asking.

Prerogative, meaning to have a right, is from Latin praerogātīvus, meaning asked first, to ask before anyone else. (prae-, pre- + rogāre, to ask.)

Derogative is from Latin dērogāre, meaning to repeal some part of a law, modify it. It is from Latin de- + rogāre to ask; sort of like asking back.

Interrogative's root in rogare is clear, to ask between (another in conversation), ask someone else.

Other words with a [prefix + rogare] as a root:

rogation (a prayer - to ask [?])
arrogate: claim for oneself; presume (to ask to/for/towards)
arrogant: proud; presuming. (ask/claim for oneself)
abrogate: do away with; annual; (ask away from)
interrogate: (obvious)
surrogate : (ask in place of, in substitution)

There are more, mostly used in Law. But you can see they are all about asking

Solution 3:

Etymology seems to comferm the 'rogare' thesis .

derogative (adj.) late 15c., from Middle French derogatif, from Latin *derogativus, from past participle stem of derogare (see derogatory).

interrogative (adj.) c.1500, from Late Latin interrogativus "pertaining to a question," from Latin interrogat-, past participle stem of interrogare "to ask, question" (see interrogation) + -ive.

prerogative (n.) "special right or privilege granted to someone," late 14c. (in Anglo-Latin from late 13c.), from Old French prerogative (14c.), Medieval Latin prerogativa "special right," from Latin praerogativa "prerogative, previous choice or election," originally (with tribus, centuria) "unit of 100 voters who by lot voted first in the Roman comita," noun use of fem. of praerogativus (adj.) "chosen to vote first," from praerogere "ask before others," from prae- "before" (see pre-) + rogare "to ask" (see rogation).

Solution 4:

There is a connection between the three words indeed and the common element is the verb "rogare". This verb means "ask for, plead", but it also means "propose a law". If you consider also this second meaning you can easily explain why the three words are connected. "Prerogativus" in latin indicates a person who votes before the others, or better, a person who "is asked to vote for first" (=praerogare). "Derogative" comes from the verb "derogate", which in its transitive form means "diparage, insult"; but as an intransitive verb "derogate" means to "take away a part so as to impair, to detract". Here what the Merriam-Webster say about the origin of DEROGATE: "Middle English, from Late Latin derogatus, past participle of derogare, from Latin, to annul (a law), detract, from de- + rogare to ask, propose (a law)"