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 Originally Posted by Poopadoop
What the highest-order derivative that has a use in the real world?
For example, in movement we sometimes measure jerk (second derivative of velocity, or third of position) as a measure of "smoothness" of a movement, but nothing beyond that. Is there anything in any domain that goes higher than that and what's it used for?
x -> position
(d/dt) x -> velocity
(d/dt)^2 x -> acceleration
(d/dt)^3 x -> jerk
(d/dt)^4 x -> snap
(d/dt)^5 x -> crackle
(d/dt)^6 x -> pop
and no, I'm not even joking about the last 3.
Off the top of my head, elasticity theory frequently uses the 4th derivative, but it's a spacial derivative, not a time derivative. I.e. (d/dx) instead of (d/dt). Elasticity theory covers how materials change shape when forces (loads) are applied. This describes how an object in bending will bend such that it minimizes the curvature for a given loading.
It also applies in civil engineering when determining how a road should bend. Minimizing the 4th derivative (w.r.t. space) makes for smooth turns.
I do not know of any branch of physics or engineering that regularly uses a higher order derivative.
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