はじめる・だす: the difference
はじめるN4
Meaning
To start, To begin to
Past papers
Tested 2 times in past papers
Example
今朝、雪が降りはじめた。
It started to snow this morning.
だすN4
Meaning
To suddenly begin, To start doing
Past papers
Tested once in past papers
Example
あの子は急に泣き出した。
That child suddenly started crying.
How to tell them apart
「～はじめる」 is a planned or calm start. 「～だす」 is a sudden start: 「雨が降りだした」 (It suddenly began to rain), 「赤ちゃんが泣きだした」 (The baby suddenly burst into tears).
For a planned start, use 「はじめる」: 「明日から勉強しはじめる」, not 「×明日から勉強しだす」.
It does not go with verbs that finish in an instant, such as 「死ぬ」, 「落ちる」, 「着く」.