# Which examples should we mention when teaching the concept of derivatives?

I am teaching Calculus for non-maths major students. As far as I know, when we teach about derivatives, we should mention "the rate of change". There are some practical examples to motivate this concept. For example, the velocity of a car moving on the street.

However, I would like to find an example that may fascinate our students a bit more. Could anyone help me with some suggestions?

• Most calculus book have such sections and exercises. For example Thomas' calculus have many examples on physics, biology and economics. Commented Jan 28, 2019 at 17:24
• Specifying a major would help to find relevant examples. A civil engineer and an anthropologist might be interested in different things. Commented Jan 28, 2019 at 17:53
• Walking up and down hills measuring height above sea level. The the movement of the needle on a car's speedometer. Commented Jan 29, 2019 at 17:21

I dont know whether this is more or less fascinating than the position/velocity/acceleration examples of derivatives with respect to time, but for a practical example of a derivative students are all used to, I ask them to put one hand on their wooden(?) desktop and grab the metal leg of the desk with the other hand.

But these have been in the room together for hours (years, probably), so surely they should be the same temperature. Thus, students are really encountering the difference in thermal conductivity of two materials, and they are experiencing heat transfer at different rates (from their hand to the table leg or the desk). The common notion of "cold" here is a derivative that we all have built-in. Most students find this example pretty interesting. We then try to come up with other examples of things we experience and quantify that are actually rates.

• I'm not too happy about calling thermal conductivity a rate of chsnge. They're experiencing a particular temperature drop at their skin surface, due to a rate of heat flow from their skin, determined by thermal conductivity of the table leg, and what's changing with time is heat energy in their body and the table leg. I feel this example is like mixing up the electrical conductivity of a material with the current flowing through it. Commented Jan 29, 2019 at 23:41
• I'd be happy with "they're experiencing a rate of heat flow" though. Commented Jan 29, 2019 at 23:42
• Perhaps I misused the phrase "thermal conductivity" in this context, though I do not say those words to my students. I do tell them they are experiencing higher or lower heat transfer, and this is a rate. Commented Jan 30, 2019 at 2:27
• That sounds accurate. I'm now trying to remember how the phenomenon was first explained to me—by specific heat capacity I think (more heat needed to warm up a metal object to skin temperature) but I think conduction is more relevant. Commented Jan 30, 2019 at 3:04
• @timtfj The higher rate of transfer is experienced as a colder hand -- it's the integral of the rate that they experience, no? (This really goes beyond my knowledge: what do the neurons actually react to?) Commented May 3, 2023 at 13:06

One of my favorite examples is to explain why the derivative of the area of a circle is the circumference. And the derivative of the volume of a sphere is the surface area. If you try the same for the square and cube, it may not work at first, but try to not use the length of the side as the parameter, but half the length of the side.

You can argue in many ways. My favorite is to cut the ring and bend it out to get something that looks like rectangle with width equal to the circumference, and height equal to $$\Delta r$$.

• Nitpick: "the derivative of the area of a circle is the circumference" is true, provided you mean the derivative with respect to the radius. The derivative of the area wrt the circumference for instance, is the radius. Wrt the the diameter it's half the circumference or wrt the area is 1 etc. Commented Apr 28, 2023 at 7:29

The typical example of a rate of change is one that changes with respect to time. I would strongly suggest introducing at least one example where the independent variable is not a quantity of time.

One relatively-easy-to-visualize example is to find the rate of change of a shape's area with respect to one of its lengths. For example, a square's area $$A$$ is the square of the length of its side $$x$$, so $$\frac{\mathrm{d}A}{\mathrm{d}x}=\frac{\mathrm{d}}{\mathrm{d}x}x^2=2x$$. Note that it is a function of $$x$$. That is, the rate of change of a square's area with respect to its side length depends on what the side length currently is.

This example easily leads to the discussion on the chain rule. If a square's side length is changing at a time rate of, say, $$2$$ meters per second, then its area is changing at a time rate of $$\frac{\mathrm{d}A}{\mathrm{d}t}=\frac{\mathrm{d}A}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d}t}=2x(2$$ m/s$$)$$ (where $$x$$ is a quantity having meters as the unit of measurement, so $$\frac{\mathrm{d}A}{\mathrm{d}t}$$ is in m$$^2$$/s). As the square gets bigger, its area is changing faster with respect to time. (That is, the time rate of change of the side length is constant, but the time rate of change of the area is not.)

To emphasize the importance of rate-of-change vs. actual value a graph of the world population can be useful.

The world population now is more than 100 years ago, but more importantly the derivative now is much much greater than 100 years ago.

Suggestion from a non-teacher. When you get on to second and third derivatives, how about a simplistic model of global warming? Assuming things haven't had time to reach equilibrium, and making various possibly wild assumptions:

• the rate of warming depends on the difference between incoming and outgoing heat radiation
• that depends on the amount of CO$$_2$$ in the atmosphere, so more CO$$_2\Rightarrow$$ faster warming, not just higher temperature (unless equilibrium is reached)
• the rate of increase of CO$$_2$$ depends on the rate at which we produce it (strictly, how much that exceeds the rate at which natural processes can absorb it)
• now relate the behaviour of the temperature to how fast the rate of emission is increasing (for example).

I think that can get you up to about the fourth derivative, with meaningful interpretations all along, and there should be a few ways to play around with it. (Unfortunately the real equations are probably hideously complicated.)

Actually I've just realised you can get yet another derivative out of it if you assume the rate of ice loss is proportional to the excess temperature.

Or maybe save all this as an integration example for later on, or something to think about qualitatively just before introducing integration properly.

I recommend to stick to kinematic motion as the initial, most common application. "How fast the car goes." It is just the easiest to understand from daily life (even if you are a nurse, not an engineer/physicist). Of course there are many other functions (thermal, financial, etc.) But a lot of these are more abstract ESPECIALLY if time is NOT the independent variable.

Remember for weaker students, don't get too tricky at first. While all the regulars here are good at math, you need to consider your audience is not them, but your actual students. Practical pedagogy--not "interesting". In particular, an easy example to stay away from is the Joel's area example--too abstract for new, weak, students. (Not to go after Joel but to make an example of what not to do.)

Finally, this question shows a lack of any initial effort. I would say this to a student asking for homework help. But a teacher? Can't you at least scan a couple textbooks? Do a 1 minute Google search? We can help you better (and you help yourself more also) if you make some small initial effort yourself.