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As we all know, if you differentiate distance with reference to time, you get speed, and likewise, differentiating speed you get acceleration. However, if you keep differentiating, to the rate of change of acceleration and so forth (more emphasis on the so forth): is there any point to this? Does it have any useful application in any industrial or practical industry, as opposed to theoretical physics?

Also, what could we possibly learn about a situation by understanding the rate of the rate of the rate of change of the objects distance?

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The amplitude of quadrupole radiation is proportional to the source's value of $d^3 x/dt^3$. This applies to both electromagnetic radiation and gravitational radiation (for which quadrupole radiation is the lowest possible multipolarity).

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