A Positive Third Derivative

In mechanics, an object has a position. Using calculus, we can take derivatives of that position with respect to time:

• The first derivative is velocity. It tells us how the position changes over time.
• The second derivative is acceleration, which describes how quickly velocity changes. With positive acceleration, an object steadily gains speed.
• The third derivative, it turns out, has a name too: jerk. A positive third derivative means that not only is velocity increasing, but acceleration is increasing as well—in other words, the object is gaining speed at an ever faster rate.

I have felt the effects of a positive third derivative several times in my life.

Long ago, I owned a marvel of Japanese engineering: an all-wheel-drive Infiniti G35x sports sedan. Its horsepower was off the charts—a full 316 thoroughbreds under the bonnet. One summer I drove it all the way to the Italian Riviera (see the second photo). I will write about that someday.

While travelling through Germany, I tried an experiment on one of its autobahns, on a stretch with no speed limit. In quick succession, I shifted the gearbox from normal to sport mode, used the left paddle to drop down a gear, and floored the accelerator.

Anyone who has flown knows how an aeroplane accelerates: rapidly, but evenly. In other words, the second derivative—acceleration—is high, while the third—jerk—is zero. The Infiniti was different: its acceleration itself kept increasing. I could feel it through my whole body. It was an incredible sensation. At moments like that, your internal gyroscope and vestibular system must be in shock.

The second time I experienced a positive third derivative was on the French Riviera, when I rented a Ferrari F430 F1 Spider. It was, of course, far more responsive and aggressive than the premium Japanese sedan, but the principle was more or less the same: acceleration was changing. Unfortunately, the novelty had already worn off. I never thought I would say that about a Ferrari ;))

Today I feel the effects of a positive third derivative in my professional life. Everything is changing very quickly, and the pace of that change is itself accelerating. My colleagues and I discussed these rapid shifts at the International Technology Congress 2026. I have already fallen into the habit of checking for updates to the tools I use several times a day. Where updates once arrived every few days, now there can be two in a single day—and each one substantially changes what the application can do.

What an extraordinary time to be alive!