Name: deriving the slope of a catenary curve
Uploaded: 2026-01-12T12:24:17.000Z
Duration: 1 min 25 s
Channel: JoVE

The shape of a suspension bridge cable, when hanging under its own weight, follows a catenary curve, modeled using the hyperbolic cosine function.When the vertical position y1 is known, the inverse hyperbolic cosine function helps find the corresponding horizontal position x1.The inverse hyperbolic functions include the inverse hyperbolic sine, cosine, and tangent, along with their cosecant, secant, and cotangent counterparts.To find the derivative of the inverse hyperbolic cosine function, first express it in terms of the hyperbolic cosine function.Differentiating both sides implicitly gives a relation involving the hyperbolic sine function.Using the standard identity that relates the hyperbolic sine and cosine, and substituting the expression for hyperbolic sine, gives the derivative in terms of the hyperbolic cosine. Eliminating the hyperbolic cosine gives the final expression in terms of the vertical position.This expression shows that the rate of change of the horizontal position of a catenary curve of a suspension bridge depends on the vertical position.The steepness is greater near the supports and smaller at the center.

inverse hyperbolic functions and their derivatives

Inverse Hyperbolic Functions and Their Derivatives

The shape of a suspension bridge cable hanging under its own weight is described by a catenary curve, which is modeled using the hyperbolic cosine function. This mathematical model accurately captures the balance between gravity and tension acting along the cable. When a particular vertical position on the cable is known, the corresponding horizontal position can be determined using the inverse hyperbolic cosine function, allowing for a detailed analysis of the cable's geometry.Inverse hyperbolic functions form a family that includes the inverse hyperbolic sine, cosine, and tangent, along with their corresponding cosecant, secant, and cotangent forms. These functions play a crucial role in calculus when solving problems that involve hyperbolic relationships, such as those encountered in structural engineering and physics.To determine how the horizontal position changes with respect to the vertical position along a catenary, the derivative of the inverse hyperbolic cosine function must be found. This process begins by rewriting the inverse function in terms of the original hyperbolic cosine function. Implicit differentiation is then applied, resulting in an expression that involves the hyperbolic sine function. By using a standard identity that relates the hyperbolic sine and cosine functions, the derivative can be rewritten entirely in terms of the hyperbolic cosine. Finally, the remaining hyperbolic expression is eliminated so that the derivative depends only on the vertical position.The resulting relationship shows that the rate at which the horizontal position changes along the cable is not constant. Instead, it depends on the height of the cable above its lowest point. The slope is relatively small near the center of the bridge, where the cable is nearly horizontal, and becomes progressively larger as the cable approaches the supports. This variation in steepness reflects the physical behavior of suspension bridge cables and highlights the usefulness of hyperbolic functions in modeling real-world structures.

The shape of a suspension bridge cable hanging under its own weight is described by a catenary curve, which is modeled using the hyperbolic cosine ...

inverse-hyperbolic-functions-and-their-derivatives

Education

JoVE Core

Calculus

Unit

Differentiation Rules

KEY TERMS AND CONCEPTS

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 cosecant, secant, and cotangent forms are also part of this family. These functions are useful in calculus for problems that involve hyperbolic relationships, including cases from structural engineering and physics.

To find how the horizontal position changes with respect to the vertical position on a catenary, the derivative of the inverse hyperbolic cosine function is needed. The derivation starts by rewriting the inverse function using the original hyperbolic cosine function. Implicit differentiation is then applied, which brings in the hyperbolic sine function.

A standard identity links the hyperbolic sine and cosine functions. Using that identity, the derivative can be rewritten entirely in terms of hyperbolic cosine. The remaining hyperbolic expression is then removed so the derivative depends only on the vertical position.

The final relationship shows that the rate of change is not constant. It depends on the height of the cable above its lowest point. The slope is small near the center of the bridge, where the cable is almost horizontal, and it becomes larger as the cable moves toward the supports. This changing steepness matches the physical behavior of suspension bridge cables and shows why hyperbolic functions are useful for real-world structures.

Deriving the Slope of a Catenary Curve

derivatives of simple functions

Slope Rules for Simple Functions

Derivatives show how fast a function changes, and they also describe slope. A derivative can tell the slope of a straight line or the slope of a tangent line to a curve at one point. In this lesson, the idea is tied to a roller coaster track, where the derivative of the horizontal position tells how steep the ride is at each spot. A flat roller coaster section can be modeled with a constant function, f(x) = c, where c is a fixed height. Its derivative is f'(x) = 0, so the slope is zero and...

the product rule

Derivative of a Changing Rectangle

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the quotient rule

Differentiating Ratios with the Quotient Rule

The quotient rule is a key calculus method for differentiating ratios of two differentiable functions. It is used when a function is written as one function divided by another, and the denominator is not zero. The rule gives a direct way to find the derivative of that ratio. The quotient rule is especially useful for rational functions, trigonometric ratios, and exponential functions. For example, it can be applied to a function in ratio form to find its derivative step by step. This makes it...

derivatives of the trigonometric functions

derivatives-of-the-trigonometric-functions

Trig Derivatives in Ferris Wheel Motion

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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

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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

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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

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derivatives of inverse trigonometric functions

derivatives-of-inverse-trigonometric-functions

Inverse Trig Derivatives in Tracking

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application of rates of change

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exponential growth

Bacterial Doubling in Exponential Growth

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related rates

Balloon Inflation and Changing Rates

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linearization and approximation

Tangent Line Estimates Near a Point

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application of linearization and approximation

application-of-linearization-and-approximation

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.

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

hyperbolic-and-inverse-hyperbolic-functions-problem-solving

Tangent Lines on a Hyperbolic Arch

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key terms and concepts

Learn how inverse hyperbolic cosine helps model a suspension bridge cable and how its derivative shows changing slope along a catenary curve.

Deriving the Slope of a Catenary Curve | 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

Deriving the Slope of a Catenary Curve

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