In real life what does it mean? I remember d1 is the slope at a given point. D2… D3… D4 jerk? What do D5 and D6 mean? I took calc 2 in uni but I cant remember and have no idea anymore.
In: Mathematics
Higher-order derivatives mostly just have their place as the derivative of the derivative. The lower-order derivatives can be interpreted as visible properties, but as the order gets larger, things become much more abstract. On an arbitrary curve, dy/dx is the slope, as you have mentioned. d²y/dx² tells you something about the curvature, but starting at d³y/dx³, it becomes much more abstract (“rate of change of curvature”? That hardly says more than “third derivative” does).
x(t) is something where a lot of derivatives can be felt. x is your location. dx/dt the change of location with respect to time – your velocity. d²x/dt² tells you how much your velocity changes, an acceleration which you immediately experience. d³x/dt³ is called jerk, and it tells you about a change in acceleration:
Think of moving through a perfectly circular curve at constant speed. The only accelration you feel on the straight is gravity. Now enter the curve. Your vehicle immediately produces a centripetal acceleration to follow the curve, which you feel as a sudden onset of centrifugal force (actually just your own inertia) slamming you against the side of the vehicle, a giant spike in jerk. If the curve eased in more gently from straight to curve, you would get a finite jerk and the same or even greater acceleration would not be as unpleasant.
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