derivative / drainage / derivation
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derivative
In mathematics, the local rate of change of a function; a central concept in differential calculus.
Im Matheunterricht lernen wir gerade, wie man die Ableitung einer Funktion bildet.
In math class, we are currently learning how to find the derivative of a function.
'Die Ableitung' (the derivation/derivative) is the noun corresponding to the verb 'ableiten' (to derive/divert). The ending '-ung' is a typical way to form a noun from a verb in German. The basic idea of 'ableiten' (to take something from somewhere else or lead it away) appears in all its meanings.
Man kann viele Adjektive von Verben ableiten.
Many adjectives can be derived from verbs.
You can almost always recognize the meaning of 'Ableitung' (derivation/derivative/drainage) from the context. Words like 'Funktion' (function) or 'berechnen' (to calculate) point to mathematics. 'Wortstamm' (word stem) or 'Silbe' (syllable) point to linguistics, and 'Rohr' (pipe) or 'Wasser' (water) to the technical field.
Die Ableitung des Wortes „glücklich“ ist nicht ganz einfach zu erklären.
The derivation of the word 'glücklich' (happy) is not very easy to explain.
In a mathematical context, 'Ableitung' (derivative) is often used with the verbs 'bilden' (to form/find), 'berechnen' (to calculate), or 'bestimmen' (to determine).
Können Sie die Ableitung dieser komplizierten Funktion bilden?
Can you find the derivative of this complicated function?
In the technical meaning, it usually refers to the removal of liquids. Typical combinations are 'die Ableitung von Regenwasser' (the drainage of rainwater) or 'die Ableitung von Abwasser' (the drainage/removal of wastewater).
Ein verstopftes Rohr behinderte die Ableitung des Schmutzwassers.
A clogged pipe obstructed the drainage of the wastewater.
integration
'Integration' and 'Ableitung' are closely related in mathematics, but they mean opposite things. The 'Ableitung' (derivative) describes how strongly a function changes at a given point. 'Integration', for example, calculates the area under a curve and is the reverse of differentiation. Learners may confuse the words because both occur in calculus and are often covered in the same class. The key point is: 'Ableitung' asks about change, while 'Integration' often asks about area or a sum.
Lena bestimmt die Ableitung, weil sie die Steigung der Funktion wissen möchte.
Lena finds the derivative because she wants to know the slope of the function.
In der Matheaufgabe brauchen wir die Integration, um die Fläche unter der Kurve zu berechnen.
In the math problem, we need integration to calculate the area under the curve.