At the shortest, a dragonfly's life-cycle from egg to death of adult is about 6 months. Some of the larger dragonflies take 6 or 7 years! Most of this time is spent in the larval form, beneath the water surface, catching other invertebrates. The small damselflies live for a couple of weeks as free-flying adults. The larger dragonflies can live for 4 months in their flying stage. In Britain, lucky Damsels seldom go more than two weeks and Dragons more than two months. Most Damsels rarely go more than a week, and Dragons two or three weeks. They die from accidents and predation, and large numbers from starvation - in poor weather neither they nor their prey can fly.
No insect has a lifespan of only one day - even mayflies (not closely related to dragonflies) live for several months underwater as larvae before emerging as winged adults. Adult mayflies may only live for a day or so as they are dedicated "breeding machines". They cannot feed as adults as (most) don't have any functional mouthparts.
http://www.dragonflysoc.org.uk/frameset.htm?fa q&faq