When I learned about derivative, integral, acceleration and speed at school, I thought about it right away - see the picture.But didn’t you think such thoughts in your head?
The derivative of acceleration is sometimes known as "jerk".
Alan
The derivative of jerk sometimes is called "jounce" or "snap"
The derivative of jounce may be called "crackle"
And the sixth derivative of position could be called "pop".
Seventh derivative is "lock"
Eighth derivative is "drop"
And what you called "way" in the last line (the integral of position) may be called "absement" or "absition".
Source: http://wearcam.org/absement/Derivatives_of_displacement.htm