How did mechanical speedometers "calculate" velocity , being given velocity is the derivative of distance? If I am correct, velocity ( instantaneous speed) is the derivative of distance. 
Nowadays cars are equipped with electronic calculators, but previously that wasn't the case. 
Hence my question : by which mechanical means did old speedometers manage to give an instantaneous rate of change ( in distance)  , as if they could " calculate" a derivative? 
Another problem: suppose a car is moving at time t, in order to calculate the velocity at t, the quotitient [s(t+h) - s(t)] / h must be defined ( for velocity is the limit of this quotient). But, at time t, the number s(t+h) is not defined ( for it is the distance the car will have travelled in some very close future instant). 
So, could one say that the velocity given at time t by a speedometer cannot be anything else but a " retrospective " velovity? 
 A: in an analog speedo as used in a car, the speedo cable (driven by a front wheel) spins a small permanent magnet shaped like a disc. Near that magnet is a little aluminum cup that encloses the disc from one side. The speedo needle is attached to that disc, which is allowed to turn against a fine hairspring as used in moving-coil ammeters and the like. 
The rotating magnet induces eddy currents in the aluminum cup, and those currents establish a magnetic field which reacts with the rotating magnet in such a manner as to drag the aluminum cup around along with it. the torque applied to the needle cup is proportional to the velocity of the spinning magnet, and so the deflection of the needle against the restoring force of the spring is proportional to the rotation speed of the magnet. 
A: First:
Original mechanical speedometers worked by connecting a magnet directly to a the drive shaft.  This would then impart a force on a metal "speed cup".  So the metal cup didn't just spin freely it due to the pull from the magnet/metal force, it was balanced with a spring.  And the difference in applied force by magnet on the cup and restorative force imparted by the spring, also on the cup, puts your dial at a certain spot.
Second:
The idea of pure instantaneous difference, or rate of change is more a mathematical concept and less a practically literal one for 
real world application.  Yes even electronic systems are reading back to you an old speed since it takes time for mechanical and/or electrical stuffs to to act and reach your speed dial.  And frankly EVERY measurement is retrospective because intantaneous information transfer is not a physical reality (barring some novel research in quantum).  But it can be faster than you'd be able to physically notice or sense it.  
Third:
Do you remember Taylor's series, and all the other ways to approximate area under a curve.  That is how calculators and computers do the all math, they do not do it symbolically.  They do not actually send a limit to infinity, per the formal derivative definition.  They set it to a high enough number that the approximation is good enough for the particular situation.
A: An answer by Jon Custer, from the comments:

They spun a little magnet to generate a voltage to move the needle. Source: disassembly of our VW Bug speedometer in the 1970s. 

Another anecdote. As the speedometer failed in my 1977 Chevrolet, there was a month or less where the rotation data was transmitted (mechanically) to the console, but not transmitted continuously. So if I idled through a parking lot at a walking speed like 5--10 mph, the speedometer would read zero, then zero, the POING up to forty or fifty, then glide down to zero, then zero, then POING!  As I sped up, the POINGs started to become less dramatic and to arrive before the needle has finished falling to zero, and the needle would wiggle around a reasonable estimate of my speed. Above twenty or thirty it looked like a normally functioning speedometer --- until one day when I heard the cable break and then it said zero at all speeds.
This kind of physical "needle averaging" is why we talk about root-mean-square amplitudes for alternating currents --- not necessarily because it's mathematically useful, but more because that's what a generation of tools could measure.
A version of the POING problem is still present in modern speedometers that count axle rotations digitally. Look closely at the speedometer of the next car you're in: many or most are nonlinear under about twenty mph, so that this can be hidden from you.
