Much of our intuition for derivatives comes from physics:
velocity (rate of change) is easily the 1st derivative;
acceleration/force is the 2nd

It's easier to see -changes- in derivatives than simply non-vanishing
(for example, it's hard to see that the exponential is -so- big -- it just
looks like it increases fast).
Perhaps this is related to needing several points to see derivatives?
You need two points to see increasing/decreasing, three points to see
convexity/concavity, so 4 points to see the next derivative...

However, since many forces are -constant-, we only get quadratics.

Further, we have a rather hard time seeing 2nd derivatives (say, zeros in
second derivative other than inflection points), and really can't see third
derivatives (which is why sketches of x^3 look bad).

For instance, most fonts are drawn with quadratics (TrueType) or cubics
(PostScript) -- there's no need for quartics as we can't see that.
Similarly cubic splines/surfaces are enough when modeling
(polygons look jagged, but cubics look smooth).

Beyond that there's some natural cases where exponentials occur, but higher
degree polynomials/derivatives are unusual/harder to interpret.

There are a few places that higher order equations arise; here's my favorite:
if you drop a penny/rock in a well, it will decrease in size -quartically-
(i.e., degree 4).
This is b/c it moves away quadratically (constant velocity) and also shrinks
quadratically in visual size with distance -- cute, eh?

If you're sitting in a car, then the force pushing you into your seat is the
acceleration, so you can equate your position with the acceleration of the
car, and can thus understand higher derivatives of speed in terms of your
position. The "jerk" is the speed at which you are being moved around in your
seat.

if there is a discontinuity in the jerk, this corresponds to
non-differentiability in the acceleration, i.e., a new force entering and
"jerking" you around (e.g., a collision: your acceleration goes from 0 to -60
or whatever)
Jerks cause damage to components and discomfort: when you drop something, it's
not the acceleration that causes the damage (when the object is falling, it's
accelerating but undamaged) -- it's the jerk at the end.

You can also understand the 4th derivative as how quickly you are
-accelerating- in your seat.
Another example is when you have a change in sign of jerk: you start moving in
the other direction in your seat.

x(t) = position
d^1x = velocity
d^2x = acceleration
d^3x = jerk (in england, called "jolt")
no standard terms for higher derivatives, as they don't come up often.

(ballpark: about g/5 per second (a fifth of gravity per second) is pretty
"smooth" feeling)