Origins
Derivative traces back to water. Latin derivare meant to lead water away from a stream or to divert a channel from a river, from de- (away from) and rivus (stream). The image is concrete and agricultural: a farmer cuts a channel from a brook to irrigate a field. The diverted water flows from the original source but serves a different purpose. Late Latin derivativus extended this to mean anything drawn from a source.
The word entered English in the 15th century and immediately found use in grammar. A derivative word is one formed from another: darkness from dark, quickly from quick, unhappy from happy. The grammatical application preserves the water metaphor perfectly — the derived word flows from its source word like a channel from a river.
Mathematics adopted derivative in the 17th century for the function that describes the rate of change of another function. Newton and Leibniz, developing calculus independently, needed vocabulary for operations that extract new mathematical objects from existing ones. Derivative served because the calculated function is derived from the original, flowing from it through a specific operation.
Later History
The financial use, now the most publicly prominent, developed in the late 20th century. A financial derivative — options, futures, swaps, and similar instruments — derives its value from an underlying asset such as a stock, bond, commodity, or interest rate. The derivative does not contain the asset itself; its value flows from the asset's performance, exactly as an irrigation channel's flow depends on the river it was cut from.
Latin rivus also produced river (through French riviere), rival (originally someone sharing the same stream), rivulet (a small stream), and arrive (to reach the shore, to come to the riverbank). All these words share the ancient image of flowing water that the farmers who first dug derivative channels would have recognized immediately.