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

Consideremos un sistema de tres marchas, donde x, y y z representan las velocidades de rotación de la primera, segunda y tercera marcha, respectivamente.La primera marcha gira la segunda, que gira la tercera. El sistema se analiza utilizando la Regla de la Cadena con una relación de variables de dos etapas.La primera etapa expresa la dependencia de la rotación de la segunda marcha respecto a la primera, con z en función de x.La segunda etapa expresa la dependencia de la rotación de la tercera marcha respecto a la segunda, con y como función de z.De ello se deduce que la salida de la primera etapa se convierte en la entrada de la siguiente, formando una función compuesta.Un pequeño cambio en una marcha provoca cambios en las otras. Para encontrar la tasa instantánea de cambio en y sobre x, se toma la razón de Delta y sobre Delta x con el límite cuando Delta x se acerca a cero. Al introducir la variable intermedia, delta z, esta razón se reescribe.Cuando Delta x se acerca a cero, Delta z también se acerca a cero. Entonces el límite del producto de razones se expresa como el producto de dos derivadas, dando lugar a la Regla de la Cadena. Esto da la derivada de una función compuesta como la derivada de la función exterior multiplicada por la derivada de la función interna.

la regla de la cadena

Un sistema de engranajes interconectados proporciona una interpretaci&#xF3;n f&#xED;sica concreta de la regla de la cadena en el c&#xE1;lculo. Considere tres engranajes dispuestos en secuencia, en la que las velocidades de rotaci&#xF3;n del primero, segundo y tercer engranajes est&#xE1;n representadas por las variables x, z e y, respectivamente. El primer engranaje impulsa al segundo, y este al tercero, por lo que el movimiento de cada engranaje depende del anterior. Esta estructura conduce naturalmente a una relaci&#xF3;n de variables de dos etapas que puede analizarse mediante funciones compuestas.Relaciones funcionales compuestasEn la primera etapa, la rotaci&#xF3;n del segundo engranaje depende de la rotaci&#xF3;n del primero, por lo que la variable intermedia se expresa como z = z(x). En la segunda etapa, la rotaci&#xF3;n del tercer engranaje depende del segundo, lo que da como resultado y = y(z). Dado que la salida de la primera etapa sirve como entrada a la segunda, la relaci&#xF3;n general entre el primer y tercer engranaje es una funci&#xF3;n compuesta, escrita como y = y(z(x)).Tasas de cambio y la regla de la cadenaUn peque&#xF1;o cambio en la velocidad de rotaci&#xF3;n del primer engranaje produce un cambio correspondiente en el segundo, que a su vez afecta al tercer engranaje. Para determinar la tasa instant&#xE1;nea de variaci&#xF3;n del tercer engranaje con respecto al primero, se considera el cociente &#x394;y/&#x394;x en el l&#xED;mite cuando &#x394;x tiende a cero. Al introducir el cambio intermedio &#x394;z, esta raz&#xF3;n se puede expresar como:\begin{equation*} \jfrac{\Delta y}{\Delta x} = \jfrac{\Delta y}{\Delta z} \jfrac{\Delta z}{\Delta x} \end{equation*}A medida que &#x394;x tiende a 0, el cambio correspondiente &#x394;z tambi&#xE9;n tiende a 0, lo que permite interpretar cada raz&#xF3;n como una derivada.Esto nos lleva directamente a la regla de la cadena, que establece que la derivada de una funci&#xF3;n compuesta es el producto de la derivada de la funci&#xF3;n externa por la derivada de la funci&#xF3;n interna:\begin{equation*} \jfrac{dy}{dx} = \jfrac{dy}{dz} \jfrac{dz}{dx} \end{equation*}En el sistema de engranajes, este resultado muestra que la tasa de cambio general entre el primer y el tercer engranaje est&#xE1; determinada por la respuesta de cada engranaje individual al que lo impulsa.

La regla de la cadena

Un sistema de engranajes interconectados proporciona una interpretaci&#xF3;n f&#xED;sica concreta de la regla de la cadena en el c&#xE1;lculo. Considere ...

Education

Educación

JoVE Core

Calculus

Cálculo

Unit

Unidad

Differentiation Rules

Reglas de derivación

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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

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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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