Physical intuition for higher order derivatives

If you want a simple intuitive explanation, you can get a lot from vehicles.

In a car traveling at a constant speed, suppose there is a white dot painted on the top of the steering wheel. If that dot is in the center, you are traveling in a straight line. If you turn it some angle to the left, say 90 degrees, then the car is traveling in a circular arc at a constant lateral acceleration. That is the second derivative of lateral position.

If you turn the steering at a constant rate from 0 degrees to 90 degrees, then the rate of lateral acceleration is changing constantly while you are turning the wheel. That is jerk, and it is constant because you are turning the steering wheel at constant speed. It is the third derivative of lateral position.

(While you are doing this, the path traced by the car is a spiral of linearly increasing curvature. Highway and railway curves are built using these spirals to connect segments of constant curvature. Such a spiral gives a place to gradually bank the roadway, and without it drivers tend to cross lanes, and trains actually jerk when starting or ending a curve.)

However, if you don't turn the steering wheel at a constant rate, but rather accelerate it leftward from 0 degrees until you are turning it quickly at 45 degrees, and then decelerate it until it reaches 90 degrees, then you are giving it a doublet of snap, first positive, then negative, and that is the fourth derivative of lateral acceleration.

Another vehicle to illustrate it is a submarine having bow planes to control depth. Suppose there is a motor that rotates the bow planes at a constant speed, up, down, or 0. The angle of the bow planes determines the pitch rate of the submarine. The pitch angle of the submarine determines the rate of change of depth.

So the pitch angle is proportional to the first derivative of depth, the bow plane angle determines the second derivative, and the speed of the bow plane motor is the third derivative.

Also, take a rocket with gimballed engines. If the thrust line of a rocket engine does not go through the rocket's center of mass, then it produces angular acceleration, or the second derivative of directional orientation. There are motors that move the engine gimbals, and the rate at which they move them determines the third derivative of direction.

I'm sure you can think of other examples.

The smoothness of acceleration and thus applied and reactive forces is related to the higher derivatives (i.e. jerk, snap, crackle and pop).

So a physical interpretation would be if you are standing in a bus, and the driver brakes with constant deceleration up to the point the bus stops at which point you feel like you are thrown backwards. Drivers to avoid this problem typically ease off the brakes before stopping to achieve a better transition. Ideally you want a 2nd order curve for acceleration, or constant snap to transition nicely into a stop.

I've always liked the discrete explanation.

In physics, you can almost always approximate a function $f: \mathbb{R} \to \mathbb{R}$ by saying what the value of $f$ is at a discrete set of points $\{x_i\} \subset \mathbb{R}$.

If you do this, you approximate the derivative by looking at how $f$ changes when you move from a point to one of its nearest neighbors. So if your mesh is a lattice of points $a\mathbb{Z} = \{an | n \in \mathbb{Z}\}$, you only get to step a distance $a$. Second order derivatives are constructed from nearest neighbor differences of derivatives, so you can travel a distance $2a$ when constructing 2nd order derivatives. Likewise, for $n$-th order derivatives, you can see information that is distance $na$ from you in the lattice.