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

Rozważmy układ trzech biegów, gdzie x, y i z reprezentują prędkości obrotowe pierwszego, drugiego i trzeciego biegu.Pierwszy bieg kręci drugim, który kręci trzecim. System analizowany jest za pomocą reguły łańcuchowej z dwustopniową relacją zmiennych.Pierwszy stopień wyraża zależność obrotu drugiego biegu od pierwszego, gdzie z jest funkcją x.Drugi stopień wyraża zależność rotacji trzeciego koła zębatego od drugiego, gdzie y jest funkcją z.Wynika z tego, że wyjście pierwszego etapu staje się wejściem następnego, tworząc funkcję złożoną.Mała zmiana jednego biegu powoduje zmiany w pozostałych. Aby znaleźć chwilową zmianę w y na x, przyjmuje się stosunek Delta y do Delta x z limitem, gdy Delta x zbliża się do zera. Wprowadzając zmienną pośrednią, delta z, ten stosunek jest przepisywany.Gdy Delta x zbliża się do zera, Delta z również zbliża się do zera. Wtedy granica iloczynu iloczynu stosunków wyraża się jako iloczyn dwóch pochodnych, co daje regułę łańcuchową. Daje to pochodną funkcji złożonej jako pochodną funkcji zewnętrznej pomnożoną przez pochodną funkcji wewnętrznej.

reguła łańcuchowa

Uk&#x142;ad po&#x142;&#x105;czonych k&#xF3;&#x142; z&#x119;batych stanowi konkretn&#x105; fizyczn&#x105; interpretacj&#x119; regu&#x142;y &#x142;a&#x144;cuchowej w rachunku r&#xF3;&#x17C;niczkowym. Rozwa&#x17C;my trzy ko&#x142;a z&#x119;bate u&#x142;o&#x17C;one sekwencyjnie, gdzie pr&#x119;dko&#x15B;ci obrotowe pierwszego, drugiego i trzeciego ko&#x142;a z&#x119;batego s&#x105; reprezentowane odpowiednio przez zmienne x, z i y. Pierwsze ko&#x142;o z&#x119;bate nap&#x119;dza drugie, a drugie nap&#x119;dza trzecie, wi&#x119;c ruch ka&#x17C;dego ko&#x142;a z&#x119;batego zale&#x17C;y od ruchu poprzedniego. Taka struktura naturalnie prowadzi do dwuetapowej zale&#x17C;no&#x15B;ci zmiennych, kt&#xF3;r&#x105; mo&#x17C;na analizowa&#x107; za pomoc&#x105; funkcji z&#x142;o&#x17C;onych.Z&#x142;o&#x17C;one zale&#x17C;no&#x15B;ci funkcyjneW pierwszym etapie obr&#xF3;t drugiego ko&#x142;a z&#x119;batego zale&#x17C;y od obrotu pierwszego ko&#x142;a z&#x119;batego, wi&#x119;c zmienna po&#x15B;rednia jest wyra&#x17C;ona jako z = z(x). W drugim etapie obr&#xF3;t trzeciego ko&#x142;a z&#x119;batego zale&#x17C;y od drugiego, co daje y = y(z). Poniewa&#x17C; wyj&#x15B;cie pierwszego stopnia s&#x142;u&#x17C;y jako wej&#x15B;cie drugiego, og&#xF3;lna zale&#x17C;no&#x15B;&#x107; mi&#x119;dzy pierwszym a trzecim ko&#x142;em z&#x119;batym jest funkcj&#x105; z&#x142;o&#x17C;on&#x105;, zapisan&#x105; jako y = y(z(x)).Szybko&#x15B;ci zmiany i regu&#x142;a &#x142;a&#x144;cuchowaNiewielka zmiana pr&#x119;dko&#x15B;ci obrotowej pierwszego ko&#x142;a z&#x119;batego powoduje proporcjonaln&#x105; zmian&#x119; pr&#x119;dko&#x15B;ci obrotowej drugiego ko&#x142;a z&#x119;batego, co z kolei wp&#x142;ywa na trzecie ko&#x142;o z&#x119;bate. Aby okre&#x15B;li&#x107; chwilow&#x105; szybko&#x15B;&#x107;, z jak&#x105; zmienia si&#x119; pr&#x119;dko&#x15B;&#x107; trzeciego ko&#x142;a z&#x119;batego wzgl&#x119;dem pierwszego, rozpatruje si&#x119; stosunek &#x394;y/&#x394;x w granicy, gdy &#x394;x zbli&#x17C;a si&#x119; do zera. Wprowadzaj&#x105;c po&#x15B;redni&#x105; zmian&#x119; &#x394;z, stosunek ten mo&#x17C;na wyrazi&#x107; jako\begin{equation*} \jfrac{\Delta y}{\Delta x} = \jfrac{\Delta y}{\Delta z} \jfrac{\Delta z}{\Delta x} \end{equation*}Gdy &#x394;x zbli&#x17C;a si&#x119; do 0, odpowiadaj&#x105;ca zmiana &#x394;z r&#xF3;wnie&#x17C; zbli&#x17C;a si&#x119; do 0, co pozwala interpretowa&#x107; ka&#x17C;dy stosunek jako pochodn&#x105;.Prowadzi to bezpo&#x15B;rednio do regu&#x142;y &#x142;a&#x144;cuchowej, kt&#xF3;ra m&#xF3;wi, &#x17C;e pochodna funkcji z&#x142;o&#x17C;onej jest iloczynem pochodnej funkcji zewn&#x119;trznej i pochodnej funkcji wewn&#x119;trznej:\begin{equation*} \jfrac{dy}{dx} = \jfrac{dy}{dz} \jfrac{dz}{dx} \end{equation*}W uk&#x142;adzie k&#xF3;&#x142; z&#x119;batych wynik ten pokazuje, &#x17C;e ca&#x142;kowita szybko&#x15B;&#x107; zmiany mi&#x119;dzy pierwszym a trzecim ko&#x142;em z&#x119;batym jest okre&#x15B;lona przez to, jak ka&#x17C;de indywidualne ko&#x142;o z&#x119;bate reaguje na ko&#x142;o z&#x119;bate, kt&#xF3;re je nap&#x119;dza.

Reguła łańcuchowa

Uk&#x142;ad po&#x142;&#x105;czonych k&#xF3;&#x142; z&#x119;batych stanowi konkretn&#x105; fizyczn&#x105; interpretacj&#x119; regu&#x142;y ...

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Differentiation Rules

Zasady różniczkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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

derivatives of simple functions

Slope Rules for Simple Functions

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the product rule

Derivative of a Changing Rectangle

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the quotient rule

Differentiating Ratios with the Quotient Rule

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derivatives of the trigonometric functions

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limits on trigonometric functions

Trigonometric Limits for Derivatives

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the chain rule

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...

the chain rule: problem solving

Chain Rule in Thermal Expansion

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implicit differentiation

Tangent Lines in Circular Motion

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second derivatives of implicit functions

Slope and Curvature of Elliptical Arches

Elliptical arches connect slope and curvature to the shape of an ellipse. In this geometry, the height y and the horizontal position x are related implicitly, so y cannot be written as an explicit function of x. That is why implicit differentiation is needed to study the arch. The ellipse is centered at the origin and has a semi-major axis a and a semi-minor axis b. Differentiating the ellipse equation with respect to x uses the chain rule. After solving for dy / dx, the first derivative shows...

implicit differentiation: problem solving

Tangent Lines with Implicit Differentiation

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derivatives of logarithmic functions

Natural Log Derivatives and Real-World Uses

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logarithmic differentiation

Differentiating Tire Stress with Logs

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the number e as a limit

e from Natural Log Derivatives

The number e is a key constant in calculus, and it helps describe continuous change. It is especially important for exponential growth. Rather than treating e as an arbitrary number, the transcript shows how it can be defined through the natural logarithm and differentiation. The natural logarithm, written as ln x, is the inverse of the exponential function with base e. Its derivative is 1 divided by its input. When that derivative is evaluated at x = 1, the result is exactly 1. This condition...

derivatives of inverse trigonometric functions

derivatives-of-inverse-trigonometric-functions

Inverse Trig Derivatives in Tracking

Inverse trigonometric derivatives can describe how a tracking angle changes as an aircraft moves closer to an observer. In this situation, a ship measures the slant distance to the aircraft and the angle of elevation. These measurements let the observer find the horizontal and vertical parts of the distance using trigonometric relationships. The angle of elevation is treated as the dependent variable, and the ratio of altitude to slant distance is treated as the independent variable. That...

application of rates of change

Velocity and Acceleration in Motion Graphs

Velocity and acceleration describe how a car moves along a highway. The car’s position changes over time, so it can be written as a function of time. Rates of change help measure motion in a clear mathematical way. Velocity tells how position changes with time. Average velocity is found by dividing the change in position by the change in time. It gives the overall speed of the car during a time interval, but it does not show changes inside that interval. To find the velocity at one exact...

exponential growth

Bacterial Doubling in Exponential Growth

Bacterial populations can grow by doubling when conditions are favorable. When nutrients and temperature support growth, cells divide by binary fission. In binary fission, one cell splits into two identical daughter cells. This makes the population rise quickly. During exponential growth, the growth rate depends on how many cells are already present. As the population gets larger, more new cells are formed in each generation. The result is a steep increase in total cell count over time. The...

related rates

Balloon Inflation and Changing Rates

Balloon inflation is a clear example of related rates, where one changing quantity affects another through a single relationship. Related rates study how linked quantities change with respect to time. As air is added, the balloon’s volume rises, and the radius must also increase. For a hot air balloon, the inflated envelope is often modeled as a perfect sphere. This simplification makes the math easier to work with. In that model, the balloon’s volume depends directly on its radius. When air...

linearization and approximation

Tangent Line Estimates Near a Point

Linearization uses a tangent line to estimate a nonlinear function near a chosen reference point. It replaces a difficult function with a simpler linear model when the input changes only a little. This makes it useful for small deviations from a known value. A square root example shows how the method works. The value at 4 is known exactly, so 4 is a good reference point. The function value and the rate of change are both easy to find there. Estimating the value at a nearby input, such as 4.1,...

application of linearization and approximation

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Estimating Altitude Changes with Linearization

Linearization can help a drone estimate small altitude changes from air pressure readings. Atmospheric pressure drops as altitude rises, and this relationship is often modeled with an exponential function. The model is accurate, but the math needed to convert pressure into altitude can be too complex for repeated calculations during flight. To make the calculation easier, the pressure-altitude relationship is approximated locally with a straight line. This line is built near a chosen reference...

hyperbolic functions

Modeling Hanging Cables with Hyperbolic Curves

Hyperbolic functions help describe the shape of a hanging cable. When a flexible cable is suspended between two points at the same height, it naturally forms a catenary. That curve comes from the balance between the cable’s weight and the tension along its length. It shows a state of mechanical equilibrium. The true shape of this hanging cable is not captured well by a simple approximation. Instead, it is described with hyperbolic functions. These functions are related to exponential functions.

inverse hyperbolic functions and their derivatives

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Deriving the Slope of a Catenary Curve

A catenary curve describes the shape of a suspension bridge cable hanging under its own weight. The curve is modeled with the hyperbolic cosine function, which helps show how gravity and tension act along the cable. When the vertical position on the cable is known, the matching horizontal position can be found with the inverse hyperbolic cosine function. Inverse hyperbolic functions are a family of functions that includes inverse hyperbolic sine, cosine, and tangent. Their corresponding...

hyperbolic and inverse hyperbolic functions: problem solving

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Tangent Lines on a Hyperbolic Arch

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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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