Name: differentiating ratios with the quotient rule
Uploaded: 2026-07-22T07:17:34.197Z
Duration: 1 min 29 s
Channel: JoVE

A water tank fills and drains simultaneously, causing the water level to change due to unequal inflow and outflow rates. The goal is to examine the ratio of water volume to drainage rate. This ratio changes over time because both quantities are differentiable functions of time. As water enters the tank, the total volume increases slightly, while the drainage rate also shifts due to system adjustments. These small changes influence the overall volume-to-drainage quotient. To find the quotient&#x27;s rate of change, consider the change in the quotient over a small time interval, and then divide this change by that interval. By introducing a limit as the time interval approaches zero, delta R also approaches zero because R varies continuously with time, simplifying the denominator. Using the definition of derivatives, the quotient rule is derived. It equals the drainage rate times the derivative of the volume, minus the volume times the derivative of the drainage rate, all divided by the square of the drainage rate. This final expression shows how the ratio of the water volume to the drainage rate changes over time. In general, this expression also relates how the ratio of two arbitrary functions changes over time.

the quotient rule

The quotient rule is a fundamental differentiation technique in calculus used to differentiate functions expressed as a ratio of two differentiable functions. Given a function of the form:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19795/19795_PT_Equation_1_editor_19795.svg" alt="19795_PT_Equation_1.svg" /></figure>Where g(x) and h(x) are both differentiable and h(x) &#x2260; 0, the derivative of f(x) is given by:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19795/19795_PT_Equation_2_editor_19795.svg" alt="19795_PT_Equation_2.svg" /></figure>Example:The quotient rule is beneficial when differentiating rational functions, trigonometric ratios, and exponential functions. For example, given:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19795/19795_PT_Equation_3_editor_19795.svg" alt="19795_PT_Equation_3.svg" /></figure>applying the quotient rule,<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19795/19795_PT_Equation_4_editor_19795.svg" alt="19795_PT_Equation_4.svg" /></figure>This rule is essential in solving problems involving rates of change in physics, engineering, and economics, making it a critical tool in mathematical analysis.

The Quotient Rule

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Education

JoVE Core

Calculus

Unit

Differentiation Rules

KEY TERMS AND CONCEPTS

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 a practical tool for solving calculus problems with fractions of functions.

The quotient rule also appears in real-world rate of change problems. It is important in physics, engineering, and economics, where changing quantities are often modeled with ratios. For this reason, it is a valuable part of mathematical analysis.

Differentiating Ratios with the Quotient Rule

differentiating-ratios-with-the-quotient-rule

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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...

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key terms and concepts

Learn how the quotient rule helps find derivatives of ratios of functions, including rational, trigonometric, and exponential examples in calculus.

Differentiating Ratios with the Quotient Rule | 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 Ratios with the Quotient Rule

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