1640s, "characterized by or appropriate to dismissal;" from dismiss + -ive. Meaning "contemptuous, tending to reject as insignificant" is recorded by 1922 (implied in dismissively). Related: Dismissiveness.
early 15c., dismissen, "release from court restraint or legal charges;" late 15c., "remove from office, service, or employment," apparently from Latin dimissus, past participle of dimittere "send away, send different ways; break up, discharge; renounce, abandon," from dis- "apart, away" (see dis-) + mittere "send, let go" (see mission). Meaning "send away, order or give permission to depart" is from 1540s.
The prefix was altered by analogy with many dis- verbs. Middle English also had dismit, in the same sense (late 14c.), with altered prefix but directly from the Latin verb, and dimit. Related: Dismissed; dismissing.
word-forming element making adjectives from verbs, meaning "pertaining to, tending to; doing, serving to do," in some cases from Old French -if, but usually directly from Latin adjectival suffix -ivus (source also of Italian and Spanish -ivo). In some words borrowed from French at an early date it has been reduced to -y (as in hasty, tardy).
<a href="https://www.etymonline.com/word/dismissive">Etymology of dismissive by etymonline</a>
Harper, D. (n.d.). Etymology of dismissive. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/dismissive