Besides worrying about higher derivatives (jerk, . . .) let's consider going
in the other direction.
Eg, physicists describe a vertical toss by the quantities:
j(t) = 0
a(t) = -g
v(t) = vo -gt
y(t) = vo t -g(t^2)/2
Each of these is the derivative (integral) of the one below(above) it.
These quantities depict what we observe, measure and find practically
useful.
But from a purely (non-applied?) mathematical viewpoint, any f(t) is a
member of a potentially infinite chain of functions, each being the
derivative /integral of its adjacent neighbors.
Eg, the above chain of free fall quantities can be extended indefinitely to:
j(t) = 0
a(t) = -g
v(t) = vo -gt
y(t) = vo t -g(t^2)/2
yy(t) = vo t^2/2 - g t^3/6
yyy(t) = vo t^3/6 - g t^4/24
. . . . ad infinitum