Learn Turkish: Tepsi nerede? Elimden kayıverdi. - Where is the tray? It suddenly slipped from my hand.

It combines the verb root kay (from kaymak, to slip) with the suddenness suffix -ıver, followed by the past tense -di. The -ıver suffix adds the meaning that the action happened very quickly or unexpectedly.

Why is the suffix -ıver used here instead of -iver?

The first vowel in the suddenness suffix follows 4-way vowel harmony. Since the root kay has the vowel "a", the suffix takes an "ı", making it -ıver. Note that the "e" in the second part of the suffix (-ver) never changes, no matter what the preceding vowel is.

Could I just say elimden kaydı instead? What's the difference?

Yes, you could. Elimden kaydı simply means "It slipped from my hand." Adding the -ıver suffix to make kayıverdi emphasizes that it happened in a flash or took you by surprise, which fits well with accidentally dropping a tray!

Why does elimden have the -den (from) ending here?

Verbs that involve leaving, dropping, or slipping away from a source take the ablative case (-dan/-den). Here, the tray is moving away "from" your hand (el + im + den), so Turkish requires this ending where English might say "slipped out of my hand."