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Something intuitive and fundamental as the concept of velocity (of a particle for example) in classical physics is defined as a derivative, a concept to me quite vague and strange, although i know its mathematical definition.

Why the need for defining the velocity using a derivative ? Can you explain this to me ?

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    $\begingroup$ What else would you like to define it as? If velocity is change in position per change in time, do you have another mathematical concept that describes that? $\endgroup$ – tpg2114 Oct 3 '19 at 11:06
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    $\begingroup$ We don't (or historically didn't) define velocity as a derivative. We defined the mathematical derivative operator in a way that describes phenomena like velocity. $\endgroup$ – The Photon Oct 3 '19 at 16:12
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Well, I might as well put my answer in too.

First, the velocity is useful because it tells us how far we travel in some time period, or in a different way if we know our velocity we can determine how long it will take to travel some distance, etc.

Now, it is very easy to work with velocity and these distance/time intervals when the rate of change is constant. For example, if I am traveling at a constant $5\,\mathrm{m/s}$, then in $3\,\mathrm s$ I travel a total distance of $(5\,\mathrm{m/s})\cdot(3\,\mathrm s)=15\,\mathrm m$. In general, for a constant velocity, we have $$\Delta x=v\Delta t$$

However, what if my velocity is changing? Well then the above relation no longer holds. Which velocity should we use? Which time interval should we use? And this is where calculus comes in. Let's say we have a changing velocity $v(t)$ and we want to know how far we travel in some time interval? Well, let's break the trip into many smaller time intervals $\text dt$. In fact, let's make $\text dt$ so small that we can essentially think of our velocity as constant (remember, constant velocity is very easy). Then, on our very small time interval we can use the above relation to determine the small distance traveled $\text dx$ during our small time interval $\text dt$ $$\text dx=v\,\text dt$$

Or, in other words, $$v=\frac{\text dx}{\text dt}$$

And this is why the derivative is so useful. Changing velocity is a hard problem to handle. Constant velocities are easy to handle. So let's just break things up into tiny pieces so we are working with constant velocities, then we can piece everything back together to get the total distance traveled. i.e. $$\Delta x=\int_{x_0}^{x_f}\text dx=\int_{t_0}^{t_f}\frac{\text dx}{\text dt}\,\text dt=\int_{t_0}^{t_f}v(t)\,\text dt$$

These ideas generalize to more than one dimension with vectors. The ideas are the same.

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You could define try to define velocity as the change in position over a finite time interval (what physicists would call average velocity). But the problem with that is the average velocity depends on the time interval chosen, so how do you choose a time interval that works for every possible motion ?

Also, average velocity has some non-intuitive properties. For example, the average velocity of a pendulum over one entire cycle is zero (because it has returned to its starting point) even though it has clearly moved during that time.

So instead physicists define velocity as the instantaneous change in position divided by time. Or, if you want to avoid vague terms like "instantaneous", you take a more mathematical approach and say velocity is the limit of the average velocity as the time interval approaches zero. In other words, velocity is the derivative of position with respect to time.

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The idea is to be able to write expressions like

$$ {\rm d}x = v \, {\rm d} t $$

which can be integrated in order to solve for particular problems. Here the velocity $v$ is treated as the tangent slope of the position-time curve. And from calculus we know that the slope of a curve is calculated using the derivative.

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A derivative is really just the rate of change of some variable against another variable.

For example think of the price of a pizza $P$ vs its radius $R$. How the price changes for a pizza when you increase its size? Well, since if you change the size by the double $R_2 = 2R_1$ the area of the pizza has increased like $A_2 = \pi R_2^2 = 4\pi R_1^2 = 4A_1$ and since we should expect the price of the pizza to go up with the amount of mass required to do it and this mass depends on the surface area of the pizza you can expect the price not to double but to quadruple (just like the surface area of the pizza).

The change of the radius variable is $\Delta R=R_2-R_1=2R_1-R_1=R_1$ and the change in the price of the pizza has been $\Delta P = P_2-P_1=4P_1$. Thus if your first pizza was of size $R_1=1$ and it costed $P=10$ (in arbitrary units) you know that the change of price was $\Delta P = 40$ when you doubled the size $\Delta R = 2$.

But a derivative is not just the change of something, a derivative is the rate of change of that thing with respect to the change in the other. In this case the derivative is $\Delta P/\Delta R=4/2=2$.

So, why velocity can be defined as some kind of derivative? Because velocity is the rate of change of the position of an object against a change of time. If you travel from $x_1$ to $x_2$ a distance of $10\; km $ between your clock times $t_1$ and $t_2$ in $8\; seconds$, then your velocity is $v=\Delta x/\Delta t=(x_2-x_1)/(t_2-t_1) = 10\;km/8\;s=1.25\;km/s$.

The only two things that one should add to this concept is:

  • The fact that velocity is not a scalar but a vector thus its dereivative is in fact the derivative of each of its components, and the rate of change ( the derivative) means that not only the magnitude of the velocity changes but also its direction in 3D space (still a rate of change but expanded to acknowledge vector quantities)
  • Derivatives are not just an average rate of change like I did but is the instantaneous rate of change, so you can describe a change in speed that might have occurred in between time $t_1$ and $t_2$ but that would have preserved the average speed identical. Those kind of details can be described thanks to the fact derivatives are not just $\Delta x/\Delta t$, where $\Delta$ represent the change of some variable, but is the limit when $\Delta x$ goes to $0$ and $\Delta t$ also goes to $0$, of that fraction; $\Delta x/\Delta t$. You want to opick smaller and smaller time intervals and smaller differences in position between those instants to know what the velocity was specifically in that short time-span and not a general average of this.
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