The speedometer in your car measures speed directly.
It works by measuring the current generated by a magnet turning in direct proportion to the turning of some component of your automobile.
The details are here. http://en.wikipedia.org/wiki/Speedometer#Eddy_current
When the car or motorcycle is in motion, a speedometer gear assembly turns a speedometer cable, which then turns the speedometer mechanism itself. A small permanent magnet affixed to the speedometer cable interacts with a small aluminum cup (called a speedcup) attached to the shaft of the pointer on the analogue speedometer instrument. As the magnet rotates near the cup, the changing magnetic field produces eddy currents in the cup, which themselves produce another magnetic field. The effect is that the magnet exerts a torque on the cup, "dragging" it, and thus the speedometer pointer, in the direction of its rotation with no mechanical connection between them.
This shows that speed can be computed directly by an analog mechanical device; and that this is a common everyday occurrence all over the world. There's no tiny little guy frantically computing derivatives inside your car's dashboard.
By the way, your question contains a logic error. You say: "That is one doesn't measure speed directly; but one must measure it indirectly." But that doesn't follow. Just because a given quantity is defined a certain way in physics, that has nothing to do with how we measure that quantity in everyday life.
After all, gravitational attraction is defined in a very complicated way by the modern theory of gravitation; yet to measure the gravitational attraction between my body and the earth, I simply step on a scale. I don't need to take a graduate level class in general relativity.