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

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.

"Which is colder?" I ask. "The table leg", they all reply.

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.

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 $\Delta r$.

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.)

Think about how social media algorithms or viral content spread. You can model the number of people who have seen a post as a function of time. The derivative here represents the instantaneous speed of that spread. At any given moment, the derivative tells you exactly how many new people are being reached per second. It is a great way to show how a small change in the initial "virality" of a post leads to a massive difference in the total audience later on. Another relatable angle is looking at personal finance, specifically how your bank balance changes based on spending habits. If you plot your total savings over a month, the derivative at any point is your "burn rate." It shows the exact speed at which you are spending or saving money at that specific moment. This is often more intuitive for students than physical motion because they can look at their own transaction history and visualize the slope of their spending curve. These examples help students realize that calculus is not just for physics labs or geometry problems. It is a tool for understanding the momentum behind things they interact with every single day. By moving away from cars and toward digital trends or personal budgets, you make the math feel like a lens for observing the real world.