Mathematics is the language of physics therefore to understand physics is to understand its language. Based on definition of derivative the smaller the change of X of difference quotient the more it approach a certain value if it exist which we call the derivative. But what does it mean in physics? If derivative is the model use to describe instantaneous velocity then it means that the smaller you observe time interval experimentally, the ratio of distance over time approaches to a certain value. But why does it approach a certain value when we observe an object moves ? What causes it to approach a constant speed in smaller interval? and why call it the instantaneous velocity/speed? The concept of derivative only describe it but have no explanation.


First consider the case where the velocity is constant so the position time graph is linear. In this case, the slope of the line represents the velocity at all points because the velocity is constant.

Now consider the case where the velocity is some arbitrary function, $v(t)$. That means that $x(t)$ is no longer linear but is some arbitrary curve as well. If we take any two points on that curve and draw a straight line between them, the slope of that line is the average velocity over the time interval between the points but the velocity might change many times throughout the interval.

$$ v_{avg} = \frac{\Delta x}{\Delta t} $$

If we make the interval sufficiently small, ie we let $\Delta t \rightarrow 0$ then we converge to the value that we call the derivative. This is then the "average" velocity over an infinitesimal interval but since the interval is infinitesimally small and we have converged, it is the instantaneous value at that point. By definition, convergence means that we will not get a better value by making the infinitesimal interval "smaller".

We usually see this represented mathematically by the expression of the limit:

$$ \lim_{h \rightarrow 0} \frac{x(t + h) - x(t)}{h} $$

Which is the formal definition of the derivative, $\frac{dx}{dt} = v$.

  • $\begingroup$ But why does it converge? What is happening in the physical world? Yes derivative describe it pretty well but the causal explanation is missing $\endgroup$ – Edz Jul 12 '18 at 16:09
  • $\begingroup$ Think of it as I describe it in terms of average velocity. If the velocity is constant than the average over any interval is the same as the instantaneous velocity. As the velocity is not generally constant, the average is just an average, but taking this average over smaller and smaller intervals eventually leads to a region where the velocity is not changing. $\endgroup$ – fhorrobin Jul 12 '18 at 16:11
  • $\begingroup$ In smaller interval velocity does not change much but why? Why in that region it approach constancy? $\endgroup$ – Edz Jul 12 '18 at 16:22
  • $\begingroup$ Because the function is well behaved enough for the derivative to exist. If the function had infinite oscillations then the derivative would not be defined. The reason for why it becomes constant comes from the fact that velocity is smooth and governed strongly only by lower order effects (acceleration, jerk). $\endgroup$ – fhorrobin Jul 12 '18 at 16:26
  • $\begingroup$ No it's the fact that there are not higher order effects that would add more oscillations. $\endgroup$ – fhorrobin Jul 12 '18 at 16:50

Imagine a car going around a racetrack at varying speeds. The definition of velocity is the displacement of the object (denoted here by x) over the time taken for such a displacement to occur (denoted by t). It can therefore be expressed in the form x/t.

Now, if we were to take t as the time taken for the car to make a complete revolution around the racetrack, thereby ending at the exact same location as it started, x = 0. The velocity of the car when we take t to be as such is therefore 0. We can see that the velocity of the car (as we have defined it) depends on when this t is taken. Or, to be more precise, it depends on what the starting t is (denoted by t0), and how long of a time interval we are using to measure its displacement over that period (denoted by △t). The initial location of the car can then be denoted as x0, and the displacement of the car over the time period △t is denoted by △x. Looking at our definition of velocity earlier, we can then say that the velocity of the car v=△x/△t, for the time interval t0 to t0 + △t.

The limit that we're taking when we discuss instantaneous velocity, is to set △t such that it approaches zero. As mentioned before, for the car going around the racetrack, the velocity v that we calculate vary greatly depending on what our △t is, for a given t0. When △t is the lap time of the car, △x=0 and v=0. As we decrease △t to a very small number, △x becomes constrained to the immediate movement of the car at t0, since the acceleration and path taken by the car after t0 matter less and less. It therefore approaches this constant value, determined by the speed and bearing of the car at t0. Any second-order behaviour of the car (acceleration, bearing drift) will have a diminishing impact on △x as we decrease △t. This explains the tending-to-a-constant effect as △t is taken to the limit.

Finally, when △t is set at the limit (infinitesimal change in t), we denote it by dt, making △x=dx. The velocity can then be described as dx/dt, over a time period t0 to t0+dt. However, because t0 is "far greater" than dt, we normally just call it dx/dt at time t0. Alternatively, one can think of it as, for any dt where the difference in behaviour of the car between t0 and t0+dt is non-negligible, one can always take a smaller dt such that it becomes negligible. The reason for the quotation marks at "far greater" is because one can argue that t0 can be set arbitrarily to 0, making the comparison moot. That is technically true, I can't think of a waterproof way of stating it oth, perhaps a "any change in t0 for which we will be concerned about"...?

As to why dx/dt at time t0 is called the instantaneous velocity at time t0, dx/dt describes the velocity of the car, at the instant t0. :)

  • $\begingroup$ Actualy I had read this same kind of explanation before but neglected it since it only explains in concrete example such as physics not in a general abstract way . Back then my head is mix up. To me, instantaneous rate of change=instantaneous velocity= derivative. Then I realize derivative is invented for the sake of describing instantaneous velocity thanks for Newton . But what about instantaneous rate of change? The general one? Is it also true that if we reduce the change of something( whatever that can possese rate of change) it also approach an instantaneous rate? $\endgroup$ – Edz Jul 12 '18 at 17:24

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