Name: derivative of a changing rectangle
Uploaded: 2026-01-12T12:24:44.000Z
Duration: 1 min 24 s
Channel: JoVE

The Product Rule differentiates functions formed by the product of two functions.It says the derivative of two functions u and v is the sum of u times the derivative of v and v times the derivative of u.To understand why, imagine resizing a rectangular window on a screen. Its area equals width times height.As both width and height change over time, the total area also changes. This change can be visualized by new pixel blocks being added along the edges and at the corner.The total area expression has three parts: one from the width, one from the height, and one from the corner. To find the rate of change, the total area change is divided by the time interval.Taking the limit as the time interval approaches zero, the first part becomes height times the rate of change of width. The second becomes width times the rate of change of height.The corner part shows the product of the tiny change in width and the tiny change in height. As the limit approaches zero, the corner part, which depends on time, vanishes.The result matches the product rule, showing each function times the derivative of the other.

the product rule

In calculus, the Product Rule provides a method for differentiating expressions that are the product of two functions. It states that the derivative of the product of two differentiable functions equals the first function times the rate of change of the second, plus the second function times the rate of change of the first.<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:193/46;" src="https://cloudfront.jove.com/files/ftp_upload/19794/19794_001_editor_19794.svg" alt="Product rule formula; calculus differentiation equation." data-altupdatedat="1775223722000" data-alt="19794_001.svg" /></figure>This rule ensures that the rate of change of the product accounts for the simultaneous variation of both functions.A compelling way to understand the Product Rule is through a geometric analogy involving a rectangle whose width and height change over time. Let the width and height represent two functions of time, making the area their product. When both dimensions vary, the change in area over a small time interval can be visualized as consisting of three components: a strip along the width, a strip along the height, and a small square at the corner.The strip along the width represents the change due to increasing width while holding height constant. The strip along the height captures the change due to increasing height while holding the width constant. The small square at the corner, representing the combined increase in both dimensions, becomes negligible as the time interval becomes very small.Dividing the total change in area by the time interval and letting that interval approach zero, the corner term becomes infinitesimally small and vanishes. What remains are the two linear contributions from the changing width and height.<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/19794/19794_002_editor_19794.svg" alt="Differentiation equation: d/dx[W(t)H(t)] method; calculus formula; derivative analysis" data-altupdatedat="1775223722000" data-alt="19794_002.svg" /></figure>This visual model reinforces the necessity of including both partial changes to capture the full dynamic behavior of the product of two changing quantities.

The Product Rule

In calculus, the Product Rule provides a method for differentiating expressions that are the product of two functions. It states that the derivative of ...

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Differentiation Rules

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derivative of a changing rectangle

The Product Rule helps find the derivative of a product of two changing functions. In calculus, it states that the derivative of one function times another equals the first function times the rate of change of the second, plus the second function times the rate of change of the first. This rule shows how to account for both functions when they vary at the same time.

A useful way to see the rule is with a rectangle whose width and height both change over time. If the width and height represent two functions of time, then the area of the rectangle is their product. As both dimensions change, the area changes in a way that reflects both directions at once.

Over a small time interval, the change in area has three parts. One strip comes from the width changing while the height stays constant. Another strip comes from the height changing while the width stays constant. A small square appears at the corner where both changes overlap.

As the time interval becomes very small, that corner square becomes negligible. When the total change in area is divided by the time interval and the interval approaches zero, the corner term vanishes. What remains are the two linear parts from the changing width and height, which matches the Product Rule.

Derivative of a Changing Rectangle

derivatives of simple functions

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The Product Rule helps find the derivative of a product of two changing functions. In calculus, it states that the derivative of one function times another equals the first function times the rate of change of the second, plus the second function times the rate of change of the first. This rule shows how to account for both functions when they vary at the same time. A useful way to see the rule is with a rectangle whose width and height both change over time. If the width and height represent...

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key terms and concepts

Learn the Product Rule through a changing rectangle model and see why the derivative of a product includes both functions’ rates of change.

Derivative of a Changing Rectangle | 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

Derivative of a Changing Rectangle

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