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

X, y ve z sırasıyla birinci, ikinci ve üçüncü viteslerin dönme hızlarını temsil ettiği üç dişli sistemini düşünün.Birinci vites ikinci vitesi döndürüyor, o da üçüncüyü döndürüyor. Sistem, iki aşamalı değişken ilişkisi olan Zincir Kuralı kullanılarak analiz edilir.Birinci aşama, ikinci vitesin dönmesinin birinci vitese bağımlılığını ifade eder ve z x'in bir fonksiyonu olarak kullanılır.İkinci aşama, üçüncü vitesin dönmesinin ikinci vitese bağımlılığını ifade eder ve y z'nin bir fonksiyonu olarak kullanılır.Sonuç olarak, birinci aşamanın çıktısı bir sonraki aşamanın girişi olur ve bileşik bir fonksiyon oluşturur.Bir vitesteki küçük bir değişiklik diğerlerinde de değişikliklere neden olur. Y'nin x üzerindeki anlık değişim hızını bulmak için, Delta x sıfıra yaklaştıkça Delta y'nin Delta x üzerindeki oranı sınırla alınır. Ara değişken olan delta z eklenerek bu oran yeniden yazılır.Delta x sıfıra yaklaşırken, Delta z de sıfıra yaklaşır. O zaman oranların çarpımının sınırı, iki türevin çarpımı olarak ifade edilir ve Zincir Kuralı ortaya çıkar. Bu, bir bileşik fonksiyonun türevini dış fonksiyonun türevi olarak iç fonksiyonun türevi ile çarpımı olarak verir.

zincirleme kuralı

Birbirine ba&#x11F;l&#x131; di&#x15F;lilerden olu&#x15F;an bir sistem, analizde Zincirleme Kural&#x131;&#x2019;n&#x131;n somut bir fiziksel yorumunu sunar. S&#x131;ral&#x131; bi&#xE7;imde d&#xFC;zenlenmi&#x15F; &#xFC;&#xE7; di&#x15F;li d&#xFC;&#x15F;&#xFC;n&#xFC;ld&#xFC;&#x11F;&#xFC;nde, birinci, ikinci ve &#xFC;&#xE7;&#xFC;nc&#xFC; di&#x15F;lilerin d&#xF6;nme h&#x131;zlar&#x131; s&#x131;ras&#x131;yla x, z ve y de&#x11F;i&#x15F;kenleriyle temsil edilir. Birinci di&#x15F;li ikinciyi, ikinci di&#x15F;li ise &#xFC;&#xE7;&#xFC;nc&#xFC;y&#xFC; harekete ge&#xE7;irir; dolay&#x131;s&#x131;yla her bir di&#x15F;linin hareketi, kendisinden &#xF6;nce gelen di&#x15F;liye ba&#x11F;l&#x131;d&#x131;r. Bu yap&#x131;, bile&#x15F;ik fonksiyonlar arac&#x131;l&#x131;&#x11F;&#x131;yla analiz edilebilen iki a&#x15F;amal&#x131; bir de&#x11F;i&#x15F;ken ili&#x15F;kisini do&#x11F;al olarak do&#x11F;urur.Bile&#x15F;ik Fonksiyonel &#x130;li&#x15F;kilerBirinci a&#x15F;amada, ikinci di&#x15F;linin d&#xF6;nme hareketi birinci di&#x15F;linin d&#xF6;nmesine ba&#x11F;l&#x131;d&#x131;r; bu nedenle ara de&#x11F;i&#x15F;ken z = z(x) bi&#xE7;iminde ifade edilir. &#x130;kinci a&#x15F;amada ise &#xFC;&#xE7;&#xFC;nc&#xFC; di&#x15F;linin d&#xF6;nme hareketi ikinci di&#x15F;liye ba&#x11F;l&#x131;d&#x131;r ve bu durum y = y(z) ili&#x15F;kisiyle g&#xF6;sterilir. Birinci a&#x15F;aman&#x131;n &#xE7;&#x131;kt&#x131;s&#x131; ikinci a&#x15F;aman&#x131;n girdisi olarak kullan&#x131;ld&#x131;&#x11F;&#x131;ndan, birinci ve &#xFC;&#xE7;&#xFC;nc&#xFC; di&#x15F;liler aras&#x131;ndaki toplam ili&#x15F;ki, y = y(z(x)) &#x15F;eklinde yaz&#x131;lan bir bile&#x15F;ik fonksiyonla ifade edilir.De&#x11F;i&#x15F;im Oranlar&#x131; ve Zincirleme Kural&#x131;Birinci di&#x15F;linin d&#xF6;nme h&#x131;z&#x131;ndaki k&#xFC;&#xE7;&#xFC;k bir de&#x11F;i&#x15F;iklik, ikinci di&#x15F;lide buna kar&#x15F;&#x131;l&#x131;k gelen bir de&#x11F;i&#x15F;ime yol a&#xE7;ar ve bu de&#x11F;i&#x15F;im de &#xFC;&#xE7;&#xFC;nc&#xFC; di&#x15F;linin hareketini etkiler. &#xDC;&#xE7;&#xFC;nc&#xFC; di&#x15F;linin h&#x131;z&#x131;n&#x131;n birinci di&#x15F;liye g&#xF6;re anl&#x131;k de&#x11F;i&#x15F;im oran&#x131;n&#x131; belirlemek i&#xE7;in, &#x394;y/&#x394;x oran&#x131;, &#x394;x s&#x131;f&#x131;ra yakla&#x15F;&#x131;rken limit al&#x131;n&#x131;r. Ara de&#x11F;i&#x15F;im &#x394;z tan&#x131;t&#x131;ld&#x131;&#x11F;&#x131;nda, bu oran &#x15F;u &#x15F;ekilde ifade edilebilir:\begin{equation*} \jfrac{\Delta y}{\Delta x} = \jfrac{\Delta y}{\Delta z} \jfrac{\Delta z}{\Delta x} \end{equation*}&#x394;x s&#x131;f&#x131;ra yakla&#x15F;t&#x131;k&#xE7;a, buna kar&#x15F;&#x131;l&#x131;k gelen de&#x11F;i&#x15F;im &#x394;z de s&#x131;f&#x131;ra yakla&#x15F;&#x131;r ve b&#xF6;ylece her bir oran bir t&#xFC;rev olarak yorumlanabilir.Bu, bile&#x15F;ik bir fonksiyonun t&#xFC;revinin d&#x131;&#x15F; fonksiyonun t&#xFC;revi ile i&#xE7; fonksiyonun t&#xFC;revinin &#xE7;arp&#x131;m&#x131; oldu&#x11F;unu belirten Zincirleme Kural&#x131;&#x2019;na do&#x11F;rudan g&#xF6;t&#xFC;r&#xFC;r:\begin{equation*} \jfrac{dy}{dx} = \jfrac{dy}{dz} \jfrac{dz}{dx} \end{equation*}Di&#x15F;li sistemi ba&#x11F;lam&#x131;nda bu sonu&#xE7;, birinci ve &#xFC;&#xE7;&#xFC;nc&#xFC; di&#x15F;liler aras&#x131;ndaki genel de&#x11F;i&#x15F;im oran&#x131;n&#x131;n, her bir di&#x15F;linin kendisini harekete ge&#xE7;iren di&#x15F;linin hareketine verdi&#x11F;i yan&#x131;ta g&#xF6;re belirlendi&#x11F;ini g&#xF6;sterir.

Zincirleme Kuralı

Birbirine ba&#x11F;l&#x131; di&#x15F;lilerden olu&#x15F;an bir sistem, analizde Zincirleme Kural&#x131;&#x2019;n&#x131;n somut bir fiziksel yorumunu ...

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Differentiation Rules

Türev Alma Kuralları

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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