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

Consider a system of three gears, where x, y, and z represent the rotational speeds of the first, second, and third gears, respectively.The first gear turns the second, which turns the third. The system is analyzed using the Chain Rule with a two-stage variable relationship.The first stage expresses the dependence of the second gear&#8217;s rotation on the first, with z as a function of x.The second stage expresses the dependence of the third gear&#8217;s rotation on the second, with y as a function of z.It follows that the output of the first stage becomes the input of the next, forming a composite function.A small change in one gear causes changes in the others. To find the instantaneous rate of change in y on x, the ratio of Delta y over Delta x is taken with the limit as Delta x approaches zero. By introducing the intermediate variable, delta z, this ratio is rewritten.As Delta x approaches zero, Delta z also approaches zero. Then the limit of the product of ratios is expressed as the product of two derivatives, giving the Chain Rule. This gives the derivative of a composite function as the derivative of the outer function multiplied by the inner function&#8217;s derivative.

the chain rule

A system of interconnected gears provides a concrete physical interpretation of the Chain Rule in calculus. Consider three gears arranged in sequence, where the rotational speeds of the first, second, and third gears are represented by the variables x, z, and y, respectively. The first gear drives the second, and the second drives the third, so the motion of each gear depends on the one preceding it. This structure naturally leads to a two-stage variable relationship that can be analyzed using composite functions.Composite Functional RelationshipsIn the first stage, the rotation of the second gear depends on the rotation of the first gear, so the intermediate variable is expressed as z = z(x). In the second stage, the rotation of the third gear depends on the second, resulting in y = y(z). Because the output of the first stage serves as the input to the second, the overall relationship between the first and third gears is a composite function, written as y = y(z(x)).Rates of Change and the Chain RuleA small change in the rotational speed of the first gear results in a corresponding change in the second gear, which in turn affects the third gear. To determine the instantaneous rate at which the third gear&#8217;s speed changes with respect to the first, the ratio &#916;y/&#916;x is considered in the limit as &#916;x approaches zero. By introducing the intermediate change &#916;z, this ratio can be expressed as<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19798/PT_19798_1_editor_19798.svg" alt="Partial derivatives equation &#916;y/&#916;x=&#916;y&#916;z/&#916;z&#916;x, mathematical concept diagram." data-altupdatedat="1775223722000" data-alt="PT_19798_1.svg" /></figure>As &#916;x approaches 0, the corresponding change &#916;z also approaches 0, allowing each ratio to be interpreted as a derivative.This leads directly to the Chain Rule, which states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function:<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19798/PT_19798_2_editor_19798.svg" alt="Differential chain rule formula, dy/dx=dy/dz&#215;dz/dx, symbolic equation representation." data-altupdatedat="1775223722000" data-alt="PT_19798_2.svg" /></figure>In the gear system, this result shows that the overall rate of change between the first and third gears is determined by how each individual gear responds to the one driving it.

The Chain Rule

A system of interconnected gears provides a concrete physical interpretation of the Chain Rule in calculus. Consider three gears arranged in sequence, ...

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Calculus

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Differentiation Rules

KEY TERMS AND CONCEPTS

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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