die Entbindung
delivery
The medical term for the process of giving birth to a child.
Die werdende Mutter wurde zur Entbindung ins Krankenhaus gebracht.
The expectant mother was taken to the hospital for delivery.

delivery / release
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delivery
The medical term for the process of giving birth to a child.
Die werdende Mutter wurde zur Entbindung ins Krankenhaus gebracht.
The expectant mother was taken to the hospital for delivery.

The noun 'Entbindung' is derived from the verb 'entbinden'. This verb literally means 'to free from being bound to something' or 'to untie', and this basic idea is present in both meanings.
Der Arzt kann Sie von Ihrer Schweigepflicht entbinden.
The doctor can release you from your duty of confidentiality.
You can almost always recognize the meaning 'release from a duty' by the preposition 'von' (from). The structure is 'die Entbindung von etwas (Dativ)' (release from something in the dative).
Er bat um die Entbindung von seinen Aufgaben als Projektleiter.
He requested release from his duties as project manager.
In the meaning 'Geburt' (birth), 'Entbindung' (delivery) is a medical and formal term. It is often found in hospitals, at doctors' offices, or in official documents.
Der voraussichtliche Termin für die Entbindung ist der 15. Mai.
The expected date of delivery is May 15.
As 'Befreiung von einer Pflicht' (release from a duty), 'Entbindung' (release) is a term from legal and administrative language. It sounds very formal and is used mainly in written or official contexts.
Für die Aussage vor Gericht ist eine Entbindung von der ärztlichen Schweigepflicht notwendig.
For testimony in court, a release from medical confidentiality is necessary.
birth
'die Geburt' and 'die Entbindung' both refer to the process in which a child is born. 'Geburt' is the normal, general word and fits almost all everyday situations. 'Entbindung' sounds more medical and is used especially in hospitals, by doctors and midwives, or in forms. Learners may confuse the words because they often describe the same situation. When talking with friends, people usually say 'Geburt'; in a medical context, 'Entbindung' is more natural.

Die Ärztin bereitete alles für die Entbindung vor.
The doctor prepared everything for the delivery.
Zur Geburt ihres Sohnes bekam Sofia viele Glückwünsche.
Sofia received many congratulations on the birth of her son.