I understand P'(t) alot easier, but I'd like to understand the other notation. (p = position, t = time, v = velocity, a = acceleration).

$\displaystyle \frac{dp(t)}{dt}$

My first question.. is this an actual fraction? What's in the numerator, and what's in the denominator?

Is the input variable necessary in the top? Could it just be:

$\displaystyle \frac{dp}{dt}$

Now, about stacking derivatives:

$\displaystyle \frac{d^2p(t)}{dt^2}$

Ok.. are these actual exponents or are they just notations to indicate how many derivatives have been done? And why is it that the "exponent" is placed on d in the numerator and placed on t in the denominator?