derive

The everyday sense keeps the shape of its Latin parent, dērīvō, from dē 'away' and rīvus 'a stream': to lead water off a river. What you derive is drawn toward you from somewhere else. That might be knowledge from books, an income from a profession, as Dickens has it, or a poem's structure from mathematics. The pattern is fixed: derive something from something. Despite the look, it is unrelated to arrive.

Here the thing itself becomes the subject and the verb takes no object. Skills derive from army service, and in Stirling Everard's account a locomotive firebox derives from the Bulleid Pacifics. It almost always travels with from. It sits very close to the passive of the 'obtain' meaning: a design derives from, or is derived from, an earlier one. Both are correct, so choose by whether the origin or the act of taking is in focus.

Logic and mathematics give the verb its strictest form. A conclusion counts as derived only when each step follows from the premises by a valid rule of inference, so the result follows necessarily from what was assumed. The construction is the familiar one, a theorem derived from axioms, but the standard is higher. A plausible guess is inferred, while a derived result must survive every step of checking.

The etymologist's working verb: one derives a word from its source, or says that a word is derived from Latin or French. The method rests on traceable forms rather than resemblance. Derive itself looks like a cousin of arrive, but the two are unrelated. Its real lineage runs through Old French to Latin, and that story is told under the water-channeling meaning.

In chemistry the verb names a transformation rather than a borrowing: the starting compound is changed by a reaction, not merely drawn on. That is what separates it from the general 'obtain' meaning. You derive comfort from a book without altering the book, but a compound derived from salicylic acid is no longer salicylic acid. The construction stays the same: derive one substance from another.

Careful writers avoid this use. Mathematicians differentiate a function, and the result is its derivative. Using derive for that operation is proscribed, partly because it collides with the reasoning meaning, where deriving a formula means working it out from premises. To say "derive the position function" is ambiguous on a page full of proofs. Differentiate says exactly which operation is meant.

The oldest picture in the word is a stream led off its course. Latin dērīvō joined dē, "away", to rīvus, "a stream", the same stream that flows through rival. Early English spelled it deriue and used it of workmen drawing water off by drains. The "obtain" meaning still carries the image of drawing something away from a source.