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Can you suggest some useful things we can do with higher order derivatives?

A fellow student in my Calculus and Analytic Geometry I class asked what some applications of higher order (e.g., ( $y^{(n)}$, $n\geq 4$) ) derivatives might be. Our instructor (masters level faculty) did not know of one. It's got me curious...

This question doesn't require a rigorous example, just illustrative concept(s).

Thanks.

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    $\begingroup$ Please note that $y^n$ means the $n$th power of $n$. To denote the $n$th derivative of $n$, one usually uses $y^{(n)}$. $\endgroup$ Oct 11, 2011 at 4:39
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    $\begingroup$ Thanks, I meant to do that. It's my first time to use the y^(n) notation. I just learned it today! $\endgroup$
    – FreeTrader
    Oct 11, 2011 at 4:42
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    $\begingroup$ For the case of some sequences, you can treat them as the values of the derivatives of the generating function of the sequence. $\endgroup$ Oct 11, 2011 at 4:45
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    $\begingroup$ You might be interested in this question as well. $\endgroup$ Oct 11, 2011 at 4:46
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    $\begingroup$ A similar question has now been asked at mathoverflow.net/questions/335039/… which may be consulted for some more examples. $\endgroup$ Jun 29, 2019 at 12:41

5 Answers 5

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If $y(t)$ denotes the position at time $t$, then:

  • The first derivative, $y'(t)$, denotes velocity at time $t$.
  • The second derivative, $y''(t)$, denotes acceleration at time $t$.
  • The third derivative, $y'''(t)$, denotes the jerk or jolt at time $t$, an important quantity in engineering and motion control
  • The fourth derivative, $y^{(4)}(t)$, denotes the jounce at time $t$; the jounce is also used in studying motion, and in studying the cosmological equation of state.

Fifth and sixth derivatives of position are also important in some applications/theoretical physics studies, but they have no universally accepted name.

You can also see this article from the Proceedings of the National Academy of Sciences discussing the use of the fifth derivative and curve fitting to do DNA analysis and population matching.

And higher derivatives are also used for approximating functions using Taylor polynomials, which can be useful when a certain amount of precision is required.

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    $\begingroup$ I've heard 4th, 5th, and 6th derivatives called "snap", "crackle", and "pop". $\endgroup$ Oct 11, 2011 at 13:51
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The Euler-Bernoulli equation, which describes the relationship between a beam's deflection and the applied load, involves a 4th derivative.

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For an everyday answer instead of a mathematical formula: when you drive a car, the position of the steering wheel and of the gas pedal determine the acceleration of the car as a whole (that is, the second derivative of the car's position). Therefore when either the steering wheel or the gas pedal undergoes acceleration, that acceleration translates into the fourth derivative of the car's position.

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  • $\begingroup$ So when I apply a force to the gas pedal or a torque to the steering wheel, I'm affecting the car's position at 4th order. No wonder a car will travel such a surprisingly long distance when you slam on the brakes in an emergency! $\endgroup$ Jun 21, 2021 at 22:33
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Sometimes we would like to to get a rough sketch of a function around some point $x=a$, but the function is very "flat" at that point - after we work out the first few derivatives we get $ f(a) = f'(a) =f''(a) = f^{(3)}(a) = f^{ (4)}(a) = 0 $, which means that, assuming finally $ f^{(5)} \neq 0 $, the function behaves like $$ \frac{ f^{(5)}(a)}{5!} (x-a)^5 $$

near that point. Hence if the $5$-th derivative is positive at $a$, the shape is approximately like that of $x^5$ at the origin, and if the derivative is negative then the shape is like that of $-x^5$ at the origin.

One example I can think of when this arises is in the study of the stability of equilibrium points of differential equations. The stablity of the equilibria is determined by the behaviour near $x=a$, which is partly determined by the sign of the first non-zero derivative evaluated at the equilibrium point.

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Another answer said:

The third derivative, $y'''(t)$, denotes the jerk or jolt at time t, an important quantity in engineering and motion control

Keep in mind that motion-control machines are controlled by difference equations that can only roughly approximate $y(t)$, $y'(t)$, $y''(t)$, and $y'''(t)$. But then this is true of any application that connects various orders of derivatives to causal processes that interface to the real world.

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