次第に details

gradually + rise

shidai ni agaru

gradually; by degrees — a change that arrives in steps rather than at once

meaning: rise gradually

最初さいしょは戸惑とまどっても、毎日まいにち使つかっていれば次第にしだいに慣れるなれる。

saisho wa tomadotte mo, mainichi tsukatte ireba shidai ni nareru.

Even if it throws you at first, use it every day and you get used to it gradually. neutral

日ひが沈しずむにつれて、辺あたりは次第しだいに暗くらくなった。

hi ga shizumu ni tsurete, atari wa shidai ni kuraku natta.

As the sun went down, the surroundings gradually grew dark. neutral

Examples

gradually; step by step — the change accumulates rather than arriving at a stroke

最初さいしょは戸惑とまどっても、毎日まいにち使つかっていれば次第しだいに慣なれる。

saisho wa tomadotte mo, mainichi tsukatte ireba shidaininareru.

Even if it throws you at first, use it every day and you get used to it gradually. neutral

日ひが沈しずむにつれて、辺あたりは次第しだいに暗くらくなった。

hi ga shizumu ni tsurete, atari wa shidai ni kuraku natta.

As the sun went down, the surroundings gradually grew dark. neutral

調査ちょうさが進すすむにつれ、事故じこの原因げんいんは次第しだいに明あきらかになった。

chousa ga susumu ni tsure, jiko no gen'in wa shidai ni akiraka ni natta.

As the investigation went on, the cause of the accident gradually became clear. neutral

おかげさまで、業績ぎょうせきも次第しだいに回復かいふくしてまいりました。

okagesama de, gyouseki mo shidai ni kaifuku shite mairimashita.

Thanks to your support, our results have gradually recovered. formal

つつある
つつある is a predicate ending that says the change is in progress and is stiffer still; 次第に is an adverb that adds the stepwise manner and can sit on a finished change (次第に暗くらくなった), which つつある, being progressive, cannot.
だんだん
The same stepwise change on opposite ground — だんだん is the spoken word and reads chatty in written work, while 次第に is the prose one and sounds stiff in conversation; formality of speech is not the axis, so だんだん survives in keigo (だんだん分わかってまいりました).
ますます
次第に describes a change unfolding in steps and says nothing about where it started; ますます asserts the trend was already running and has stepped up, so it needs a prior state the listener already knows about.

次第
Two kanji shared, nothing else — 〜次第 rides a ます-stem and fires the moment the first act finishes (届とどき次第), while 次第に stands free in front of the predicate and describes a change with no trigger and no finishing line.
次第で
〜次第だ／次第で follows a NOUN and makes the outcome depend on it (天気てんき次第だ); 次第に modifies a predicate and makes no outcome depend on anything, so only 次第で can be answered with "depends on what?".

くなる
An i-adjective has no verb of its own to modify, so 次第に reaches it through くなる — 次第に暑あつくなる, never ×次第に暑あつい as a plain assertion.