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Depois vs. após

In Portuguese, both depois and após translate to "after" in English. However, they are not always interchangeable. The main difference lies in their formality and commonality, with depois being far more frequent and versatile in everyday language, while após is reserved for more formal or specific contexts.

Depois

A1
The word depois is the most common and versatile term for "after" or "afterwards". It is used in all contexts, from very informal to formal. It is often followed by the preposition de (forming depois de) when it precedes a noun or an infinitive verb. It can also stand alone as an adverb meaning "later" or "then".
Vamos ao cinema depois do jantar.
(Let's go to the movies after dinner.)
Eu ligo para você depois de chegar em casa.
(I'll call you after I get home.)
Primeiro termine suas tarefas, e depois pode jogar.
(First finish your chores, and then you can play.)
Ele saiu e voltou duas horas depois.
(He left and came back two hours later.)
O que aconteceu depois que eu saí?
(What happened after I left?)

Após

B1
The word após is a more formal equivalent of "after". It is primarily used in written language, such as journalism, literature, and academic texts, as well as in formal speech. It always comes directly before a noun and is never used with the preposition de. It can sometimes imply a more immediate succession of events than depois. In some contexts, it can also mean "behind" in the sense of pursuit.
O jogador deu uma entrevista após a partida.
(The player gave an interview after the match.)
A equipe de resgate chegou ao local minutos após o acidente.
(The rescue team arrived at the scene minutes after the accident.)
Dia após dia, ele estudava para o exame.
(Day after day, he studied for the exam.)
O suspeito foi capturado após uma longa perseguição.
(The suspect was captured after a long chase.)
O cachorro correu após a bola.
(The dog ran after the ball.)

Summary

In essence, depois is the standard, all-purpose word for "after" suitable for any situation, and it's often used as depois de. In contrast, após is a more formal and less common alternative, used mainly in writing and formal contexts directly before a noun. For everyday conversation, depois is almost always the correct and most natural choice.