Question

difference between 1st and 2nd derivatives of deterministic Newtonian equation?

Answer 1

Newtonian physics (accelaration * mass = force, acceleration is a second derivative)

The list goes on and on, but in general the most common and most useful equations in physics are second order, and the trend carries on in other fields as well.

Why is this the case? My guess is that nature works according to extremely complex rules requiring any and all derivatives if they are even applicable, and it our own human limitations which lead us to think that second order is good enough.

In terms of what the first and second derivatives mean, they are rates of change that tell you how quickly a function changes given a change in its input. The second derivative is just the derivative of the first derivative; you apply the derivatives in a chain. And you can do this as many times as the derivatives are defined, getting 3rd derivatives, 4th derivatives, etc.

Integration goes in the opposite direction, undoing the derivative you applied. Which leads to a whole chain of other functions going in the opposite direction as the derivatives, telling you

e.g. how far you've traveled.