Geometrical representation of the nth derivative

[AlbertEinstein]

The first derivative represents the slope of a tangent at a point on the function's curve.
The second derivative represents the concavity of the function's curve.
However I am unable to figure out what the other derivatives of a function represent either physically or geometrically.
Pleae help.

 

[quasar987]

Well if the 2nd derivative measures the speed at which the slope is changing , the 3rd derivative measures the speed of the speed at which the slope is changing, i.e. the acceleration of the slope.

That's as deep as I can get. I don't know what that says about the geometry of the curve.

 

[StatusX]

One thing I can tell you is that while knowing a finite number of derivatives of a curve at a point gives you local properties of the curve, knowing the derivatives of all orders tells you global properties via Taylor's theorem. For example, when physicists want a local theory, they cannot use operators that involve derivatives of all orders.

 

[cracker]

Hmm... did not know that there was a 3rd dirivative. Guess your learn something new every day!

 

[hypermorphism]

Sometimes the second derivative yields no information about concavity. Consider the two different functions f(x) = x^4 and g(x) = -x^4. These two functions have different concavities, but if evaluated at (0, 0), their second derivative is the same. A higher derivative will reveal their different orientations at (0, 0).
Physically, the name "jerk" is given to the third derivative of a position function with respect to time. There are standardized limits placed on jerk for things as simple as starting and stopping trains. Limits are placed on much higher derivatives for sensitive instruments like the Hubble telescope. Very few people have deemed it necessary to give names to these quantities beyond fourth derivative, fifth derivative and so on.