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If the first derivative helps us to know whether the function is in increment or in decrement

The second one helps us to know wether the function is concave upward or downward

So what about the third derviative?! What does it help us to know

To this moment I don't have any guesses

same with the fourth derivative

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marked as duplicate by Michael Hoppe, amd, Gibbs, José Carlos Santos, max_zorn Jul 7 '18 at 0:28

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  • $\begingroup$ in physics, derivative shows how fast a function changes, so you could keep applying this: if a particle's movement is described by $s$, then $s'$ describes the velocity, $s''$ describes acceleration, $s'''$ describes how fast the acceleration changes etc etc. $\endgroup$ – Alvin Lepik Jul 6 '18 at 11:06
  • $\begingroup$ en.wikipedia.org/wiki/Jerk_(physics) $\endgroup$ – Arthur Jul 6 '18 at 11:11
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    $\begingroup$ @AlvinLepik According to Wikipedia, 'how fast the acceleration changes' is called Jerk, and the derivative of jerk is Jounce. $\endgroup$ – CiaPan Jul 6 '18 at 11:20
  • $\begingroup$ See math.stackexchange.com/questions/14841/… $\endgroup$ – amd Jul 6 '18 at 17:54
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The first non-zero derivative at a point tells you the approximate behaviour near to it. For example, $x^3$ has a stationary point of inflection at $0$, and near $0$ the second derivative had the same sign as $x$, whence the first derivative is minimal at $x=0$, making this point neither a maximum nor a minimum. By contrast, the least derivative of $x^4$ that doesn't vanish at $x=0$ is the fourth, so the same logic shows the second derivative is non-negative in the neighborhood, the first derivative has the same sign as $x$, and the function is minimised at $x=0$. It is the parity of the number of times we have to differentiate to get a non-zero result that matters, not the number itself.

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