(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.

**Answer**

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

tanδ=13ϱdϱdϕ=dydx−1+(dydx)23(d2ydx2)2d3ydx3

where ϱ is the radius of curvature and ϕ is the tangential angle.

From these considerations, one could derive an expression for the osculating parabola: given a curve represented parametrically as (f(t)g(t))T, the parametric equations for the osculating parabola of the curve at t=t0 are

(f(t0)g(t0))+ϱcos4δ2(cosϕ−sinϕsinϕcosϕ)⋅((u2−2)tanδ−2u−tan3δ(u+tanδ)2)

Here for instance is the cardioid (2cost+cos2t2sint+sin2t)T and its osculating parabola at t=2π/3:

and an animation of the various osculating parabolas for the curve (3cost−2cos3t3sint−2sin3t)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.

**Attribution***Source : Link , Question Author : Gil Kalai , Answer Author : J. M. ain’t a mathematician*