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I need to know the meaning of the higher order derivatives of a polynomial.

Let make an example. Assume we have a polynomial of degree n. Then $$ f(x)=a_0+a_1x+a_2x^2+\ldots+a_nx^n $$

We know that the first derivative means the slope of the polynomial (in a certain point, x) and also we know that the second derivative means the curvature of the polynomial (again at the certain point). So, I need to know if other derivatives have any meaning for us.

BigM
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TPArrow
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    You can also consider the second derivative to be the gradient of the first derivative; the third to be the gradient of the second etc. – Matthew Cassell Feb 24 '15 at 14:31
  • Thanks, but is there any more understandable way? – TPArrow Feb 24 '15 at 14:34
  • I don't know what you mean by "more understandable way". What is more understandable? Do you mean do higher order derivatives provide you with new information about your original curve? – Matthew Cassell Feb 24 '15 at 14:35
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    The third derivative often is referred to as the 'jerk'. That is the change in acceleration. It might also have something to do with its behavior at parties... – Joel Feb 24 '15 at 14:35
  • The third derivative and onwards may not have that much meaning or use apart from what Mr. Blatter said, but they can be used to describe the first and second derivative functions and so on. For example, if you think about it from a kinematics point of view, the 3rd and 4th derivatives of displacement can be used to describe the acceleration function. – Airdish Jan 04 '16 at 04:31

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The higher derivatives of a polynomial (or of any function $x\mapsto f(x)$, for that matter) don't have an intuitive geometrical or physical meaning. In fact, computing $f^{(6)}(3)$ is a heavy intrusion into the intimate sphere of $f$. But the values of these derivatives at some point $x_0\in{\rm dom}(f)$ are of utmost usefulness when we want to compute $f$ in the neighborhood of $x_0$ with high accuracy, and $f$ is a polynomial of degree $1000$, or real analytic, i.e., representable as a power series, in the neighborhood of $x_0$. In this case we have, according to Taylor's theorem, $$f(x)=\sum_{k=0}^n {f^{(k)}(x_0)\over k!}(x-x_0)^k+ {f^{(n+1)}(\xi)\over(n+1)!}(x-x_0)^{n+1}$$ for some $\xi$ between $x_0$ and $x$.

Christian Blatter
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First derivative: slop, velocity

Second derivative: Acceleration, Convexity or Concavity, Also slope of the first derivative

Third derivative or more: look at them in abstract way or partially (derivative of derivative). They are usually used for Taylor series or numerical calculations or to solve an equation with no physical meaning (they are used as a tool rather than being an aim).

Arashium
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If you think of a function as a quantity changing over time, the derivative is the "speed" of that quantity, it's how fast the quantity is changing at a particular instant.

The geometrical interpretation of this fact leads you to see that the first derivative is slope and the second derivative is curvature. After that, however, it's hard to get a simple visual interpretation. Nevertheless, the third derivative can still be thought of as how fast the curvature is changing, the fourth derivative as how fast the rate of change of the curvature is changing is changing, and so on. There may not be a simple visual interpretation, but there is a simple conceptual interpretation.

Jack M
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