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Apressar-se vs. despachar-se

While both apressar-se and despachar-se translate to to hurry, they possess distinct nuances regarding the intent of the action and regional usage. Apressar-se refers strictly to increasing speed due to a lack of time, whereas despachar-se often implies efficiency, getting ready, or finishing a task quickly so one can move on.

Apressar-se

B1
This verb is derived from pressa (haste/hurry) and is the most direct way to express the act of rushing. You use apressar-se when you simply need to increase the velocity of your movement or action because time is running out. It is universally understood in all Portuguese variations.
Temos de nos apressar ou vamos perder o autocarro.
(We have to hurry or we will miss the bus.)
Ela apressou-se a limpar a casa antes de as visitas chegarem.
(She hurried to clean the house before the guests arrived.)
Não precisas de te apressar, pois temos muito tempo.
(You do not need to hurry, as we have a lot of time.)
O médico apressou-se para chegar à sala de emergência.
(The doctor hurried to get to the emergency room.)

Despachar-se

B2
This verb is closely linked to getting ready or wrapping things up. While it means to hurry, the focus is on finishing the current activity (like getting dressed, eating, or working) so that you are released to go somewhere else. It corresponds to get a move on or get ready quickly. Note that this usage is extremely common in European Portuguese, whereas it is less common in Brazil.
Despacha-te, senão chegamos atrasados ao cinema!
(Hurry up (get ready), otherwise we will be late for the cinema!)
Vou tomar um duche e despacho-me em cinco minutos.
(I'm just going to take a shower and I will be ready in five minutes.)
As crianças demoraram muito a despachar-se esta manhã.
(The children took a long time to get ready this morning.)
Se te despachares com o trabalho, podemos sair mais cedo.
(If you hurry up with the work, we can leave earlier.)

Summary

In short, use apressar-se when the focus is purely on speed and moving faster against the clock. Use despachar-se when the focus is on efficiency—completing a preparation routine or task quickly so that you can proceed to the next event.