Name: chain rule in a gear system
Uploaded: 2026-01-12T12:24:29.000Z
Duration: 1 min 30 s
Channel: JoVE

Betrachten wir ein System aus drei Zahnrädern, wobei x, y und z die Drehzahlen des ersten, zweiten und dritten Zahnrads darstellen.Der erste Gang dreht den zweiten, der wiederum den dritten dreht. Das System wird mittels der Kettenregel mit einer zweistufigen Variablenbeziehung analysiert.Die erste Stufe drückt die Abhängigkeit der Rotation des zweiten Zahnrads vom ersten aus, wobei z eine Funktion von x ist.Die zweite Stufe drückt die Abhängigkeit der Rotation des dritten Zahnrads vom zweiten aus, wobei y eine Funktion von z ist.Daraus folgt, dass der Ausgang der ersten Stufe zum Eingang der nächsten wird und so eine zusammengesetzte Funktion bildet.Ein kleiner Wechsel in einem Gang führt zu Veränderungen in den anderen. Um die momentane Änderungsrate in y auf x zu bestimmen, wird das Verhältnis von Delta y zu Delta x mit dem Grenzwert genommen, wenn Delta x sich Null nähert. Durch Einführung der Zwischenvariablen Delta z wird dieses Verhältnis umgeschrieben.Wenn Delta x sich Null nähert, nähert sich auch Delta z Null. Dann wird der Grenzwert des Produkts der Verhältnisse als Produkt zweier Ableitungen ausgedrückt, was die Kettenregel ergibt. Dies ergibt die Ableitung einer zusammengesetzten Funktion als Ableitung der äußeren Funktion multipliziert mit der Ableitung der inneren Funktion.

die kettenregel

Ein System miteinander verbundener Zahnr&#xE4;der bietet eine konkrete physikalische Interpretation der Kettenregel in der Analysis. Man betrachte drei nacheinander angeordnete Zahnr&#xE4;der, deren Drehzahlen durch die Variablen x, z und y dargestellt werden. Das erste Zahnrad treibt das zweite an, und das zweite das dritte, sodass die Bewegung jedes Zahnrads vom vorhergehenden abh&#xE4;ngt. Diese Struktur f&#xFC;hrt auf nat&#xFC;rliche Weise zu einer zweistufigen Variablenbeziehung, die mithilfe von zusammengesetzten Funktionen analysiert werden kann.Zusammengesetzte funktionale BeziehungenIn der ersten Stufe h&#xE4;ngt die Drehung des zweiten Zahnrads von der Drehung des ersten ab, weshalb die Zwischenvariable als z = z(x) beschrieben wird. In der zweiten Stufe h&#xE4;ngt die Drehung des dritten Zahnrads von der des zweiten ab, woraus sich y = y(z) ergibt. Da die Ausgabe der ersten Stufe als Eingabe f&#xFC;r die zweite dient, ergibt sich f&#xFC;r die Gesamtbeziehung zwischen dem ersten und dritten Zahnrad eine zusammengesetzte Funktion, dargestellt durch y = y(z(x)).&#xC4;nderungsraten und die KettenregelEine geringe &#xC4;nderung der Drehzahl des ersten Zahnrads bewirkt eine entsprechende &#xC4;nderung des zweiten Zahnrads, welche wiederum das dritte Zahnrad beeinflusst. Um die momentane &#xC4;nderungsrate der Drehzahl des dritten Zahnrads in Abh&#xE4;ngigkeit vom ersten zu bestimmen, wird das Verh&#xE4;ltnis &#x394;y/&#x394;x im Grenzwert f&#xFC;r &#x394;x &#x2192; 0 betrachtet. Durch die Einf&#xFC;hrung der Zwischenvariable &#x394;z l&#xE4;sst sich dieses Verh&#xE4;ltnis wie folgt darstellen:\begin{equation*} \jfrac{\Delta y}{\Delta x} = \jfrac{\Delta y}{\Delta z} \jfrac{\Delta z}{\Delta x} \end{equation*}Wenn &#x394;x gegen 0 geht, n&#xE4;hert sich auch die entsprechende &#xC4;nderung &#x394;z 0, sodass jedes Verh&#xE4;ltnis als Ableitung interpretiert werden kann.Dies f&#xFC;hrt direkt zur Kettenregel, die besagt, dass die Ableitung einer zusammengesetzten Funktion das Produkt der Ableitung der &#xE4;u&#xDF;eren Funktion mit der Ableitung der inneren Funktion ist:\begin{equation*} \jfrac{dy}{dx} = \jfrac{dy}{dz} \jfrac{dz}{dx} \end{equation*}Im Zahnradsystem zeigt dieses Ergebnis, dass die Gesamt&#xE4;nderungsrate zwischen den ersten und dritten Zahnr&#xE4;dern davon bestimmt wird, wie jedes einzelne Zahnrad auf das Zahnrad reagiert, das es antreibt.

Die Kettenregel

Ein System miteinander verbundener Zahnr&#xE4;der bietet eine konkrete physikalische Interpretation der Kettenregel in der Analysis. Man betrachte drei ...

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Calculus

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Differentiation Rules

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KEY TERMS AND CONCEPTS

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The Chain Rule

chain rule in a gear system

The Chain Rule appears in a gear system with three connected gears. The first gear is represented by x, the second by z, and the third by y. Each gear’s motion depends on the gear before it, so the system creates a linked change from one stage to the next.

This setup shows a composite function, which is a function built from another function. In the first stage, the second gear’s rotation depends on the first gear, so the intermediate variable is written as z = z(x). In the second stage, the third gear depends on the second gear, so y = y(z). Because the output of the first stage becomes the input of the second, the full relationship can be written as y = y(z(x)).

The Chain Rule explains how to find the rate at which the third gear changes with respect to the first gear. A small change in x causes a change in z, and that change in z affects y. To measure this instantaneous rate, the ratio Δy/Δx is taken as Δx approaches 0, and an intermediate change Δz is used to connect the two steps.

As Δx approaches 0, the corresponding Δz also approaches 0. That lets each ratio be treated as a derivative. The result is the Chain Rule: the derivative of a composite function equals the product of the derivative of the outer function and the derivative of the inner function. In the gear system, this shows that the total rate of change from the first gear to the third gear depends on how each gear responds to the one driving it.

Chain Rule in a Gear System

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The Chain Rule appears in a gear system with three connected gears. The first gear is represented by x, the second by z, and the third by y. Each gear’s motion depends on the gear before it, so the system creates a linked change from one stage to the next. This setup shows a composite function, which is a function built from another function. In the first stage, the second gear’s rotation depends on the first gear, so the intermediate variable is written as z = z(x). In the second stage, the...

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key terms and concepts

See how three linked gears illustrate the Chain Rule, composite functions, and how a change in one gear affects the next in calculus.

Chain Rule in a Gear System | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Chain Rule in a Gear System

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