In a way that shows gratitude or recognition of value; with understanding and thankfulness.
From 'appreciative' plus the adverb suffix '-ly,' which converts adjectives to adverbs describing how an action is performed. Standard English word formation dating to the 1700s.
The '-ly' ending is doing interesting work here—it transforms a static feeling (appreciative) into a dynamic behavior (appreciatively), showing how you actually demonstrate gratitude through your actions.
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