ability
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ability
The mental or physical talent or learned skill to do something specific.
Sie besitzt die seltene Fähigkeit, Menschen allein durch ihre Stimme zu beruhigen.
She possesses the rare ability to calm people with her voice alone.
The noun 'Fähigkeit' (ability) is derived directly from the adjective 'fähig' (capable / able). The ending '-keit' is a typical way to form nouns from adjectives that describe a quality.
Er ist fähig, schwere Kisten zu tragen. Er hat die Fähigkeit dazu.
He is capable of carrying heavy boxes. He has the ability to do it.
To describe the exact ability, you usually use a construction with 'zu' (to). Either an infinitive follows ('die Fähigkeit, zu lesen' = the ability to read) or a noun in the dative ('die Fähigkeit zur Empathie' = the ability for empathy / capacity for empathy).
Er hat die Fähigkeit, komplexe Probleme schnell zu lösen.
He has the ability to solve complex problems quickly.
Common verbs used with 'Fähigkeit' (ability) are 'haben' (to have), 'besitzen' (to possess), 'erwerben' (to acquire), 'entwickeln' (to develop), and 'verlieren' (to lose).
Durch viel Übung hat er neue Fähigkeiten erworben.
Through a lot of practice, he has acquired new abilities.
art
'die Kunst' and 'die Fähigkeit' can both express that someone can do something well. 'die Fähigkeit' is neutral and general here: you can have many mental or physical abilities. 'die Kunst' sounds stronger and more special. It is often used when something is difficult, requires a lot of practice, or demands special skill. In practical terms: every 'Kunst' in this sense is a 'Fähigkeit', but not every 'Fähigkeit' is called 'Kunst'.
Mein Kollege hat die Fähigkeit, komplizierte Probleme einfach zu erklären.
My colleague has the ability to explain complicated problems simply.
Die Kunst besteht darin, ruhig zu bleiben, wenn alle gleichzeitig Fragen stellen.
The art lies in staying calm when everyone asks questions at the same time.