I'm not sure exactly how to phrase this question, but here it goes:
$v=\dfrac{dx}{dt}$ therefore $x=x_0+vt$
UNLESS there's an acceleration, in which case
$a=\dfrac{dv}{dt}$ therefore $x=x_0+v_0t+\dfrac{1}{2}at^2$
UNLESS there's a jerk, in which case
$j=\dfrac{da}{dt}$ therefore $x=x_0+v_0t+\dfrac{1}{2}a_0t^2+\dfrac{1}{6}jt^3$
Are you picking up on the pattern? Velocity is the first derivative of position with respect to time, acceleration is the second, jerk is the third, and the formula just gets longer and longer.
Let's say that, hypothetically, and object was moving such that $\dfrac{d^{500}x}{dt^{500}}$ was a constant greater than zero. Is there some formula for an object's position that implements an infinite summation of time derivatives of position? maybe following the form
$$x=\sum_{n=0}^{\infty}\dfrac{1}{n!}\dfrac{d^nx}{dt^n}t^n$$