Naively is the adverb from naive: it describes doing or believing something in a way that shows you trusted too much or expected too much, because you did not know how the world really works. It almost always attaches to a verb of thinking, believing, assuming or hoping — I naively assumed, we naively believed, she naively thought. That is by far the most common shape, and a very useful one: it is how English speakers admit their own past mistake without sounding too hard on themselves. Saying "I naively assumed the shop would be open" means "looking back, I should have checked, but I wasn't being stupid — I just didn't know". Because of that, it is often used about oneself and about the past. It can also describe the manner of an action (she smiled naively at the stranger), though that is less common. Like naive, it may be written naively or naïvely; both are correct. Note the spelling: naive keeps its e and simply adds -ly, so there is an i and an e together before the l.