derivation
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derivation
The logical, step-by-step derivation of a formula, theorem, or conclusion from existing assumptions or principles. This process makes it understandable how a result comes about and is often used in mathematics, logic, and the natural sciences.
Der Professor erklärte die Herleitung der physikalischen Formel an der Tafel.
The professor explained the derivation of the physical formula on the board.
'Herleitung' (derivation) is the nominalization of the verb 'herleiten' (to derive). This separable verb consists of 'her' (movement from a place toward the speaker) and 'leiten' (to lead). The basic idea is therefore to logically 'lead something out' from an origin, or derive it.
Kannst du mir herleiten, wie du zu diesem Ergebnis gekommen bist?
Can you derive for me how you arrived at this result?
You can recognize the meaning from the context. If it is about formulas, proofs, mathematics, or logic, it means scientific derivation. If it is about words, names, or language, it means the historical linguistic origin (etymology).
Die Herleitung dieser physikalischen Theorie ist sehr komplex.
The derivation of this physical theory is very complex.
To indicate the origin of the derivation, the preposition 'aus' (from) is almost always used with the dative, less commonly 'von' (from/of). The structure is: die Herleitung von etwas aus etwas (the derivation of something from something).
Die Herleitung des Ergebnisses aus den ursprünglichen Daten war fehlerhaft.
The derivation of the result from the original data was flawed.
In a scientific context, 'Herleitung' (derivation) is often combined with verbs such as 'nachvollziehen' (to follow/understand), 'verstehen' (to understand), or 'zeigen' (to show). Typical adjectives are 'logisch' (logical), 'mathematisch' (mathematical), or 'formal' (formal).
Ich kann die logische Herleitung dieses Beweises nicht ganz nachvollziehen.
I can't quite follow the logical derivation of this proof.
proof / evidence
'der Beweis' and 'die Herleitung' can both show why a result is correct. A 'Beweis' is meant to clearly prove in the end that a statement is true. A 'Herleitung' mainly shows the path to get there: step by step, it explains how you get from known rules or assumptions to a formula or conclusion. In mathematics and the natural sciences, the words can be close in meaning. But they are not always the same: a Herleitung makes a result understandable, while a Beweis is meant to truly secure or prove it. Learners may confuse them when both appear in exercises or technical texts.
Im Unterricht erklärt die Lehrerin die Herleitung der Formel Schritt für Schritt.
In class, the teacher explains the derivation of the formula step by step.
Der Beweis zeigt, dass die Aussage für alle Zahlen gilt.
The proof shows that the statement applies to all numbers.