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If the first derivative is the tangent to a curve, what's the geometrical interpretation of taking the second, third, ... , nth derivative?

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Usually, in the practice, the second derivative is called "curvature" and it is related with the change in direction of the tangent. Think the function describing the curve as a path. Each point is moving along that curve. Then, in a quite simplistic way, the first derivative represents the velocity while the second is the "change in direction" or the "steering" on the curve.

Now, being more precise, after the second derivative there is no actual easy phisical meaning. Each derivative can be seen as the rate of change of the previous quantity with respect to the variable under derivation.

In this sense, the 4th derivative of a function with respect to a cartesian direction represents the rate of change of the 3rd derivative in such direction.

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    $\begingroup$ To be precise (and somewhat pedantic as well :-) ), the "curvature" of the curve $y=f(x)$ is not given by $f''(x)$. The correct formula is the following: $$k=\frac{f''(x)}{\left( 1+ (f'(x))^2\right)^{\frac{3}{2}}}.$$See Wikipedia. $\endgroup$ – Giuseppe Negro Sep 30 '14 at 10:40
  • $\begingroup$ Anyway, I agree that the number $f''(x_0)$ essentially describes the curvature of the curve $y=f(x)$ at $x=x_0$. $\endgroup$ – Giuseppe Negro Sep 30 '14 at 10:41
  • $\begingroup$ Correct. Thanks for the addendum! $\endgroup$ – L. B. Sep 30 '14 at 12:06
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If you plot the gradient of the tangent for each point on the original curve, you get a second plot.

The second plot gives the rate of change of the first plot.

The second derivative is the tangent at any point on your second plot's curve, this value is the rate of change of the rate of change in your first curve.

Now repeat the process with your second plot, you now have a third plot, and the gradient of this plot tells you the rate of change of the rate of change of the rate of change of your original plot.

You can continue to repeat this process as long as the function continues to be derivable.

In more physical terms:

  • If plot 1 is the position of an object, then...
  • Plot 2 is the rate of change of the position of the object, i.e. velocity
  • Plot 3 is the rate of change of the rate of change of the object, i.e. rate of change of velocity, or acceleration
  • Plot 4 would be the rate of change of your acceleration
  • Plot 5 would be the rate of change of the rate of change of acceleration
  • Etc.
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