次第に【しだいに】

次第に means "gradually": a change that builds up over time rather than happening at once. It is an adverb in front of a verb of change – なる, かわる変わる, ふえる増える, へる減る – and often comes with てくる or ていく.

てくる brings the change up to now, and ていく carries it on from a point in the past or from now into the future.

The verb has to describe a change. An action repeated many times, such as eating or reading, takes something like すこしずつ少しずつ instead.

• × ケーキを次第に食べた → ケーキを少しずつ食べた

次第に or だんだん

だんだん means the same and is the word for everyday conversation. 次第に belongs to writing, news and narration, and sounds formal when spoken.

Without に, しだい次第 is a different pattern: "as soon as" after a ます-stem, and "depending on" after a noun.