# What is the meaning of the third derivative of a function at a point

(Originally asked on MO by AJAY.)

What is the geometric, physical, or other meaning of the third derivative of a function at a point?

If you have interesting things to say about the meaning of the first and second derivatives, please do so.

I’ve found the time, so I deleted one of my original comments in the OP and decided to expand it into a full answer.

One way to geometrically interpret the third derivative is in the notion of the osculating parabola. In much the same way that the first derivative enters into the defining equation for the tangent line (the line that best approximates your curve in the vicinity of a given point), and that the second derivative is involved in the expression for the osculating circle (the circle that best approximates your curve in the vicinity of a given point), the third derivative is required for expressing the osculating parabola, which is the parabola that best approximates… oh, you catch on quick. 😉

More specifically, if you remember the fact that four points uniquely determine a parabola, you can think of the osculating parabola as the limiting case of the parabola through four neighboring points of a given curve when those four points coalesce, or come together. The so-called aberrancy (a translation of the French “déviation“) is the tangent of the angle the axis of the osculating parabola makes with the normal line, and is given by the formula

where $\varrho$ is the radius of curvature and $\phi$ is the tangential angle.

From these considerations, one could derive an expression for the osculating parabola: given a curve represented parametrically as $(f(t)\quad g(t))^T$, the parametric equations for the osculating parabola of the curve at $t=t_0$ are

Here for instance is the cardioid $(2\cos\,t+\cos\,2t\quad 2\sin\,t+\sin\,2t)^T$ and its osculating parabola at $t=2\pi/3$:

and an animation of the various osculating parabolas for the curve $(3\cos\,t-2\cos\,3t\quad 3\sin\,t-2\sin\,3t)^T$:

Further, one could also give a geometric interpretation for the fourth derivative; what one now considers is the osculating conic (the limiting conic through five neighboring points of a curve when those five points coalesce), and one could classify points of a plane curve as elliptic, parabolic or hyperbolic depending on the nature of the osculating conic. In this respect, the discriminant of the osculating conic depends on the first four derivatives.

A lot more information is in these two articles by Steven Schot (who also wrote a nice article on the “jerk”), and the references therein.