Proving d<r>/dt = <dr/dt>: A Case Study

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The discussion centers on proving that the time derivative of the running average of a function, d<r>/dt, equals the average of the time derivative, <dr/dt>, for a sinusoidal oscillation. The running average is defined as <r>(t)/dt = 1/2Pi * Integrate[r(t)*dt,{r,t-Pi,t+Pi}]. Participants suggest using Leibniz's rule and exploring the symmetry of the integration interval to facilitate the proof. There is a sense of urgency as the original poster is approaching a homework deadline and seeks assistance in finding a solution. The conversation emphasizes the mathematical principles involved in differentiating and averaging functions in oscillatory contexts.
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Let <r> (t)/dt= 1/2Pi * Integrate[r(t)*dt,{r,t-Pi,t+Pi}] denote the running average of r over one cycle of the sinosoidal oscillation.

I have to show that d<r>/dt = <dr/dt>, it does not matter whether we differentiate or time-average first.


Should I work with Leibniz rule? Can I use some symmetry of interval, I don't really know, something I tried to do but not successfully.

Can anybody, please, show solution?
 
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