Proving Energy Conservation & Loss in Odd/Even Derivatives

[Winzer]

Homework Statement

Prove:
Even derivatives conserve energy and that odd derivatives give an energy loss.

This is for general forces acting on a system of some sort. Like drag force, etc.

Homework Equations

The Attempt at a Solution

I have no clue. I need resources. Any clues on where to begin?

 

[bdforbes]

Is there any other context given for this problem? It seems a bit vague. What are these derivatives with respect to?

 

[Winzer]

The derivatives are rates of displacement. Taken with respect to time.
An example is:
\vec{F}=-b \frac{dx}{dt} The force of drag on a particle creates an energy loss in the system.

 

[bdforbes]

I can't think of how to prove it formally but it will probably involve looking potential energy and closed loop integrations, are you familiar with those concepts?