What does the third derivative signify?
The derivative of A with respect to B tells you the rate at which A changes when B changes.
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The second derivative is the derivative of the derivative: the rate of change of the rate of change.
The third derivative is the derivative of the derivative of the derivative: the rate of change of the rate of change of the rate of change.
The further significance of this depends on what A and B are.
If you are thinking of A as the height of a curve, with B as the x-axis, then the first derivative is the slope of the curve (rate of change of the height of the curve). The second derivative is the rate of change of the slope, or the curvature. If the curve is curving upwards, like a smile, there’s a positive second derivative; if it’s curving downwards like a frown, there"s a negative second derivative; where the curve is a straight line, the second derivative is zero.
In that case the third derivative is the rate of change of the curviness. I don’t know another name for it, but if the curve goes from being concave downwards to being concave upwards, you can conclude the third derivative is positive.
On the other hand, if A is position and B is time, then the derivative of A with respect to B is velocity. The second derivative is the rate of change of velocity, or acceleration. The third derivative, the rate of change of acceleration, is called jerk.
When your car is not accelerating, you’re not being pushed back in your seat at all. When your car is accelerating mightily, you"re pushed against the back of your seat. The faster the transition between these two states, the higher the jerk.
In my experience, at least 99% of all uses of the word “jerk” (with this meaning) occur in sentences of the form, “Hey, did you know that the third time derivative of position is called jerk?” The remaining one percent of the time, the word is used seriously by automobile safety engineers and the like.
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The fourth and higher time derivatives of position are not used often enough for there to be a serious established word for them. Snap has been proposed for the fourth derivative, naturally followed by crackle and pop for the fifth and sixth derivatives.