Author Jonathan David "The Author" False Reality Publishing/Press LLC
Author Jonathan David
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Introduction to Related Rates in Differential Calculus | Calculus 1 Problem Solving [SAMPLE Lesson]

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Related Rates in Differential Calculus, Mathematical Communication, and STEM Research

This PLEM Academy research session moves between professional mathematical communication and differential calculus, with a focus on related rates. The larger lesson is how to turn a mathematical problem into an organized, repeatable algorithm while developing the software, research, and communication skills that eventually belong on a STEM resume.

Microsoft Word, LaTeX, TeXShop, and Mathematical Communication

Before beginning the calculus portion of the session, we looked at an important distinction between mathematical code and the software used to produce a mathematical document.

LaTeX is a mathematical typesetting language. It can be used in several different environments. Microsoft Word can interpret mathematical input inside its equation editor. WordPress can display mathematical notation through Jetpack LaTeX. A program such as TeXShop can process a complete LaTeX document for formal publishing.

The important distinction:

LaTeX is the language. Microsoft Word, WordPress, and TeXShop are different environments through which mathematical material can be created or displayed.

For ordinary undergraduate work, professional communication, and many workplace situations, becoming fluent with Microsoft Office can be extremely useful. A full LaTeX publishing environment becomes especially valuable when producing a formal textbook, lengthy research document, or publication requiring precise control over document structure.

The goal is not to become loyal to one piece of software. The goal is to understand what each tool is for and become capable of communicating technical information efficiently.

Mathematics Should Be Written Like a Technical Argument

One of the themes of this session is that a mathematical solution is more than a final number.

Think of the problem as a short technical essay:

  • The problem establishes what is being investigated.
  • The setup identifies the necessary equations, variables, and conditions.
  • The calculation forms the body of the argument.
  • The verification checks whether the result satisfies the mathematical conditions.
  • The final statement summarizes the result with appropriate units and notation.

This is why simply obtaining an answer is not the same as understanding mathematics.

Related Rates Are Really Algorithm Problems

Related rates often appear difficult because the wording changes dramatically from one question to another. One problem may involve a cube, another a sphere, another a ladder, and another a person walking away from a streetlight.

Underneath the wording, however, the same general process keeps appearing.

General Related Rates Algorithm

  1. Identify what quantities are changing.
  2. Write down all given values and rates.
  3. Identify the equation connecting the variables.
  4. Differentiate implicitly with respect to time.
  5. Substitute the known values only after differentiating.
  6. Solve for the requested rate.
  7. State the result with appropriate units.

The point is to develop a process that can be repeated instead of treating every new problem as an unrelated puzzle.

Related Rates Example 1: Expanding Cube

Consider a cube whose edge length is x. The volume is

\displaystyle V=x^3

If both the volume and edge length are changing with time, differentiate implicitly with respect to t:

\displaystyle \frac{dV}{dt} = \frac{d}{dt}\left(x^3\right)

Applying the chain rule gives

\displaystyle \frac{dV}{dt} = 3x^2\frac{dx}{dt}.

This small example contains the central idea behind related rates: an equation involving changing quantities becomes an equation involving their rates of change.

Why the Chain Rule Appears

When we write

\displaystyle V=x^3,

the variable x is itself changing with time. More formally, we can think of it as

\displaystyle V(t)=\left[x(t)\right]^3.

That is why differentiation with respect to time produces

\displaystyle \frac{dV}{dt} = 3x^2\frac{dx}{dt}.

Related rates are therefore an important application of implicit differentiation and the chain rule.

Related Rates Example 2: Surface Area of an Expanding Sphere

Suppose the radius of a spherical ball increases at

\displaystyle \frac{dr}{dt}=2\ \text{cm/min}.

We want to determine how quickly its surface area changes when

\displaystyle r=8\ \text{cm}.

The surface area of a sphere is

\displaystyle A=4\pi r^2.

Differentiate implicitly with respect to time:

\displaystyle \frac{dA}{dt} = 8\pi r\frac{dr}{dt}.

Now substitute the known values:

\displaystyle \frac{dA}{dt} = 8\pi(8)(2).

Therefore,

\displaystyle \frac{dA}{dt} = 128\pi\ \text{cm}^2/\text{min}.
Rate of Change of Surface Area: \displaystyle 128\pi\ \text{cm}^2/\text{min}

An Interesting Connection Between Volume and Surface Area

The volume of a sphere is

\displaystyle V=\frac{4}{3}\pi r^3.

Differentiate with respect to the radius:

\displaystyle \frac{dV}{dr} = 4\pi r^2.

But

\displaystyle A=4\pi r^2.

Therefore,

\displaystyle \frac{dV}{dr}=A.

This is exactly the kind of observation worth pausing over during research. Rather than simply applying a memorized formula, ask why a derivative produces another familiar geometric quantity and what that tells us about the relationship between radius, surface area, and volume.

When Should You Draw a Picture?

Diagrams can be helpful, but they should serve the mathematics rather than replace it.

For something as direct as the expanding sphere problem, a diagram may add very little. The equation already contains everything needed.

More complicated geometry-based related rates questions can benefit significantly from a carefully labeled diagram.

The streetlight-and-shadow problem is a good example.

Related Rates Example 3: Streetlight and Moving Shadow

Consider a 15-foot streetlight. A six-foot person walks directly away from the pole at five feet per second. We want to determine how quickly the tip of the person’s shadow is moving.

Let:

  • x = distance from the pole to the person
  • y = length of the person’s shadow
  • x + y = distance from the pole to the tip of the shadow

The important geometric observation is that the two right triangles are similar.

Therefore,

\displaystyle \frac{15}{x+y} = \frac{6}{y}.

Cross multiply:

\displaystyle 15y=6(x+y).

Simplify:

\displaystyle 15y=6x+6y
\displaystyle 9y=6x
\displaystyle y=\frac23x.

Differentiate with respect to time:

\displaystyle \frac{dy}{dt} = \frac23\frac{dx}{dt}.

Since

\displaystyle \frac{dx}{dt}=5\ \text{ft/s},

the shadow itself is increasing at

\displaystyle \frac{dy}{dt} = \frac23(5) = \frac{10}{3}\ \text{ft/s}.

The Tip of the Shadow Has Its Own Rate

The question does not ask only how quickly the shadow length changes.

The tip of the shadow is located at

\displaystyle x+y.

Therefore, its velocity is

\displaystyle \frac{d}{dt}(x+y) = \frac{dx}{dt} + \frac{dy}{dt}.

Substitute the two rates:

\displaystyle \frac{d}{dt}(x+y) = 5+\frac{10}{3}.

Thus,

\displaystyle \frac{d}{dt}(x+y) = \frac{25}{3}\ \text{ft/s}.
Speed of the Tip of the Shadow
\displaystyle \frac{25}{3}\ \text{ft/s}

Why This Problem Is Harder

The person is moving, the shadow is changing length, and the tip of the shadow is moving as a consequence of both.

This means several quantities are changing simultaneously.

The key is not memorizing this particular streetlight problem. The key is recognizing the structure:

  1. Identify the geometry.
  2. Create the relationship using similar triangles.
  3. Differentiate the relationship.
  4. Determine the shadow’s rate.
  5. Combine that rate with the person’s rate to obtain the rate of the shadow’s tip.

Once that structure is understood, sister problems involving ladders, cones, water tanks, circles, shadows, and expanding geometric objects become variations of a familiar process.

Build an Algorithm Instead of Memorizing a Solution

A useful way to think about mathematics is to compare it to programming.

Solving one Sudoku puzzle is different from writing a program capable of solving every valid Sudoku puzzle.

In the same way, memorizing the solution to one related rates problem is less valuable than understanding a process capable of handling many different related rates problems.

A stronger STEM question is:

What repeatable mathematical process would allow me to solve every problem belonging to this class of problems?

Use the Textbook as a Research Tool

During this session, even basic formulas were checked against the textbook rather than simply relying on memory or immediately searching the internet.

That is deliberate practice.

A textbook contains definitions, notation conventions, reference tables, approximations, examples, and assumptions that belong specifically to the course you are taking.

Becoming comfortable navigating those resources is part of learning how technical research works.

It can also reveal something else important: textbooks contain mistakes.

During this session, a reference in one edition of the calculus text appeared to point toward material that was no longer present in that edition. This is a reminder that even professionally published material should be read critically.

Research Participation Means Reviewing the Work

The research position does not end when the live session ends.

Research participants can review the resulting lesson, PDF, mathematical notation, and written explanations for errors, inconsistencies, or opportunities for clarification.

A student can prepare an original response in Microsoft Word, then use AI as a formatting assistant to convert that original writing into the HTML and Jetpack LaTeX structure required for the website.

This process develops several skills at the same time:

  • Attention to mathematical detail
  • Technical writing
  • Microsoft Word
  • Mathematical typesetting
  • HTML
  • WordPress
  • Responsible AI-assisted formatting
  • Research communication and peer review

Prepare for Exams Strategically

Related rates and optimization are often among the calculus topics students find most difficult because the questions require interpretation before differentiation even begins.

A useful study strategy is to pay attention to the types of problems your professor assigns.

If your homework contains a related rates problem involving a ladder and a wall, the exam may contain another changing right-triangle problem. If your assignments emphasize changing volumes, expect to understand how to construct equations involving those geometric quantities.

Do not attempt to memorize every question in the exercise bank. Learn the families of algorithms behind them.

The Bigger Lesson

Related rates are not really about cubes, spheres, streetlights, shadows, or ladders.

They are an introduction to mathematical modeling.

You begin with words. You identify variables. You establish relationships. You translate those relationships into equations. You differentiate. You interpret the result.

That same general process continues into differential equations, numerical methods, engineering, physics, computer science, and professional research.

Build the Process

Learn the notation, respect the units, identify the relationships, develop the algorithm, verify the result, and communicate the work professionally. The more consistently you practice that process, the less unfamiliar advanced STEM work becomes.

Join the PLEM Academy Research Position

The PLEM Academy Research Position is designed for students who want experience beyond simply watching solutions. Participants can take part in live mathematics, physics, engineering, textbook, and research sessions while developing professional communication and technical skills.

Research participants can review developing lessons and publications, identify typos and mathematical issues, practice scientific communication through Microsoft Word, HTML, WordPress, Jetpack LaTeX, and AI-assisted formatting, and contribute to developing research and textbook projects.

Depending on the membership option, members may also receive monthly or quarterly mailers, developing books, educational material, and additional bonus resources.

Readers interested in Jonathan David’s fiction and independent publishing projects can also explore the Fiction Fan Club for books, mailers, handmade material, and additional member bonuses.

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