Educe, evoke, elicit, extract, extort mean to bring or draw out what is hidden, latent, or reserved.
Educe usually implies the development and outward manifestation of something potential or latent.
Evoke basically suggests the voice or the words of a magician compelling spirits to leave the other world or the dead to arise from their graves.
In current use the term ordinarily implies the operation of a powerful agency that produces an effect instantly or that serves as a stimulus in arousing an emotion, a passion, or an interest.
Elicit usually implies pains, trouble, or skill in drawing something forth or out; it often implies resistance either in the person or thing that is the object of effort.
Extract implies the action of a force (as pressure or suction).
Extort implies a wringing or wresting especially from one who is reluctant or resisting.