I would appreciate a really good answer to this. Somehow my brain is not up to it.
Basically, as far as I can tell, all real rates are average rates. Every observed change takes place in a time and the magnitude of the change divided by the magnitude of the time gives us the average rate. In an instant, there is no change and the magnitude of the time is zero (or possibly infinitesimal, if you're of that bent). Instantaneity is the result of our imaginations inventing models that are logically consistent. They are fictional, in a sense, but more like historical novels, they sometimes approximate reality.
The reason instantaneous rates are important is that, because of the Mean Value Theorem, derivatives are good approximations to average rates over small intervals of time. And derivatives are easier to think with than finite differences. I don't say they're easier to compute with. That may be true, sometimes. They usefulness is in how easy they are to think about, as well as in their effectiveness at approximating the changes we observe.
The real application to me is how we use them to think about problems. This has led to advances in mathematics. Try to think of all the times the MVT is used. I'm preparing to teach numerical analysis, and it is amazing to see how much of what we understand about effective numerical approximation is based on the MVT. This is real work, done by real people, to develop methods that help others solve problems within specified tolerances.
In other words, the actual real applications are theoretical. And to understand the world and set up its problems in a form to be solved, you need a workable mathematical model. When I was a young person, this was what attracted me into science and engineering, that the mathematics I was learning was giving me a grasp of the world that would allow me to solve some of its problems.
There are many applications of derivatives. In some, an equation with derivatives is set up to be integrated. In others, something, such as energy, is to be minimized. Sometimes, the derivative of the potential function indicates the force on the system. As alluded to before, the derivative may be used to help bound the error from discretization or truncation of continuous processes. In applications outside mathematics, the calculus problem has usually been solved in the form of an algebraic formula to be applied.
Persons who merely carry out the work of calculating the solution, filling out the reports and so forth, they just need good training in their specific tasks. Sometimes you get the training on the job; sometimes you enroll in a certificate program. The ones who need to solve problems need to understand the principles and to be able to set the problem up. One concern is that when you do anything, whether you have the first or second type of job, you're likely to run into problems now and then. If you've worked anywhere for a length of time, you've probably run into people who had trouble solving problems outside their ordinary tasks.
To circle back again to my point. The real application is to use derivatives to think about things. Many things are subject to mathematical formulations. They may be in mathematics. They may be in the sciences. Derivatives may be used to solve problems and advance our understanding in mathematics and the sciences. Or they be used to understand mathematical principles in the sciences, which are just problems that other people have solved.
An example, taken from the real world, but without formulas and so not quite like what the OP seeks: It should be clear to one who knows Newton's second law, force equals the mass times the acceleration $x''(t)$, that turning off the water is potentially dangerous. It never seems to be, but that's because of building codes; so one might be forgiven for never thinking so. A whole column of water, stretching back to who know where accelerates when the taps are opened. That momentum has to be stopped when the tap is turned off. Turn it off quickly and the water has to stop quickly. So the acceleration $x''(t)$ has be large, and thus the force has to be large. So the pipes have to be strong (enough).
I was talking with a civil engineer who had built dams in Iran in the 1950s. He said he never had used calculus on the job. But I was thinking that turning off a valve on a dam must be quite a dangerous thing. He told me about one young engineer who he had told to go shut off a valve. He soon saw the pressure gauge at the station shoot up and get pegged at max. Then it shot down and was pegged at zero. Then up. Finally it went down and stayed down. Then he knew. He went to the window and saw a geyser shooting up from the busted pipe.
Another example: When you're in a tight turn in a car and the driver suddenly straightens out the steering wheel, your body tends to be thrown toward the other side of the car. Why, when the acceleration in the car has changed to zero? It's because of the tension in the muscles in your torso that are exerting a force to supply the centripetal acceleration to your body to keep it in the same trajectory as the car. When the car straightens out, the tension in your body continues the centripetal acceleration, until you relax. So if the third derivative is too great, the acceleration changes faster than your body can react, and your torso tends to follow the extension of the curve that car used to be tracking.
It's much easier to think in these terms than in terms of finite differences (average rates).
