
das Bedürfnis
need
The wish or desire for something that you need or consider important.
Nach der langen Wanderung hatte ich das dringende Bedürfnis nach einer Dusche.
After the long hike, I had an urgent need for a shower.
need
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need
The wish or desire for something that you need or consider important.
Nach der langen Wanderung hatte ich das dringende Bedürfnis nach einer Dusche.
After the long hike, I had an urgent need for a shower.

to relieve oneself
A polite euphemism for going to the toilet.
Wir müssen kurz anhalten, damit der Hund sein Bedürfnis verrichten kann.
We need to stop briefly so the dog can relieve itself.
'Das Bedürfnis' (need) is a noun derived from the increasingly old-fashioned verb 'bedürfen' (to need something). The ending '-nis' often forms nouns from verbs or adjectives that describe a state or a result.
Jeder Mensch hat Grundbedürfnisse wie Essen und Schlafen.
Every person has basic needs such as eating and sleeping.
The wish or desire is usually expressed with the preposition 'nach' and the dative. The typical construction is: 'das Bedürfnis nach etwas haben' (to have a need for something).
Nach dem langen Flug hatte sie ein starkes Bedürfnis nach Schlaf.
After the long flight, she had a strong need for sleep.
Common verbs used with 'Bedürfnis' (need) are 'ein Bedürfnis haben' (to have a need), 'ein Bedürfnis verspüren' (to feel a need), or 'ein Bedürfnis befriedigen' (to satisfy a need).
Er verspürte das Bedürfnis, mit jemandem zu reden.
He felt the need to talk to someone.
The phrase 'sein Bedürfnis verrichten' (to relieve oneself) is a euphemism, meaning a polite and indirect way of saying 'auf die Toilette gehen' (to go to the toilet). It is often used in more formal contexts or when talking about animals and small children.
Gibt es hier in der Nähe eine Möglichkeit, sein Bedürfnis zu verrichten?
Is there somewhere nearby where one can relieve oneself?
wish
Wunsch and Bedürfnis can both express a desire. A Wunsch refers to what someone would like to have or experience. A Bedürfnis feels more like a personally felt need for something important, such as peace and quiet or security.

Nach der langen Sitzung spürt Lea ein starkes Bedürfnis nach Ruhe.
After the long meeting, Lea felt a strong need for peace and quiet.
Nach der langen Sitzung hat Lea den Wunsch nach etwas Ruhe.
After the long meeting, Lea wanted some peace and quiet.