Name: differentiating tire stress with logs
Uploaded: 2026-01-12T12:24:15.000Z
Duration: 1 min 28 s
Channel: JoVE

車の重量と駆動力がタイヤにかかると、外部荷重がかかります。タイヤゴムは内部の力によってこの荷重に抵抗します。この内部抵抗は応力と呼ばれます。応力によりタイヤゴムの形状が変わり、この変形はひずみとして測定されます。ゴムにおける応力とひずみの非線形関係は、理想化された条件下で発展した数学的近似によって与えられ、Gはせん断率を示し、これは材料の変形抵抗性を特徴づけます。応力は、ひずみの関数として次の式に従って表すことができます。この変化する関係を分析するために、微分化が用いられます。基底と指数の両方に変数が含まれている場合、標準的な微分規則の適用が困難になります。対数微分は方程式を再構成することでこの過程を簡素化します。自然対数を取ることで式をより小さな項に分解し、微分を扱いやすくなります。積則と連鎖律を用いて両側を区別することで、ひずみによる応力の変化を捉えます。元の関数を微分形式に戻すことで、タイヤの応力変化率が得られます。

対数微分法

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Education

JoVE Core

Calculus

微積分学

Unit

単位

Differentiation Rules

微分法（導関数の公式）

KEY TERMS AND CONCEPTS

重要用語と概念

Logarithmic Differentiation

differentiating tire stress with logs

Logarithmic differentiation is used to find how tire stress changes with strain. In this setting, a car’s weight and driving forces act as an external load on the rubber. The rubber resists that load through internal forces called stress. The deformation that results is called strain.

The stress-strain relationship shows how the tire changes shape under load. For rubber, this relationship is nonlinear, so it does not behave like a linear elastic material. The model used here includes material parameters that describe resistance to deformation. It also uses stress and strain in both the base and the exponent, which helps represent the large deformations typical of rubber.

To study how stress evolves as strain increases, the stress-strain equation must be differentiated with respect to strain. Direct differentiation is difficult because the variable appears in more than one role. Logarithmic differentiation solves this by first taking the natural logarithm of the expression. That step breaks the function into simpler parts.

After the expression is transformed, the product rule and chain rule are used to differentiate it. The result gives the rate of change of stress with respect to strain. Substituting the original stress function back into the result gives a compact equation for how the tire rubber’s internal resistance changes as it deforms. This mathematical form helps analyze tire behavior under applied loads.

Differentiating Tire Stress with Logs

derivatives of simple functions

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the product rule

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the quotient rule

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derivatives of the trigonometric functions

derivatives-of-the-trigonometric-functions

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limits on trigonometric functions

Trigonometric Limits for Derivatives

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the chain rule

Chain Rule in a Gear System

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the chain rule: problem solving

Chain Rule in Thermal Expansion

The Chain Rule appears in thermal expansion when a rod’s length changes because its temperature changes over time. In this example, a metal rod is heated, and its length depends on temperature. The temperature itself changes with time in a quadratic way. For linear thermal expansion, the rod’s length L depends on temperature T with a constant rate of change, using the initial length L0 = 2 m and the coefficient of thermal expansion α. The temperature follows a quadratic relationship in time,...

implicit differentiation

Tangent Lines in Circular Motion

Tangent lines in circular motion show how implicit differentiation helps describe an object’s direction of motion. In classical mechanics, motion is often written as a relationship between position and time. A car moving along a straight highway with constant acceleration is a simple example where velocity is an explicit function of time, so basic differentiation works directly. A satellite in circular orbit is different because its path is given by an implicit function. The circle equation...

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

Logarithmic differentiation is used to find how tire stress changes with strain. In this setting, a car’s weight and driving forces act as an external load on the rubber. The rubber resists that load through internal forces called stress. The deformation that results is called strain. The stress-strain relationship shows how the tire changes shape under load. For rubber, this relationship is nonlinear, so it does not behave like a linear elastic material. The model used here includes material...

the number e as a limit

e from Natural Log Derivatives

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derivatives of inverse trigonometric functions

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Inverse Trig Derivatives in Tracking

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application of rates of change

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exponential growth

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related rates

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linearization and approximation

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application of linearization and approximation

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Estimating Altitude Changes with Linearization

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hyperbolic functions

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inverse hyperbolic functions and their derivatives

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Tangent Lines on a Hyperbolic Arch

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key terms and concepts

See how logarithmic differentiation is used to model tire stress and strain in rubber, and how the method simplifies a nonlinear equation.

Differentiating Tire Stress with Logs | 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

Differentiating Tire Stress with Logs

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