Interpretation of the Derivative

Date: 2/2/96 at 21:5:21
From: Anonymous
Subject: derivatives/physics/time

I was asking a friend to explain calculus to me and he said that
since I have a basic knowledge of physics (algebra-based), I also 
know calculus. He said that if you have a distance, then taking its
derivative gives you velocity and that taking the derivative of a 
velocity gives an acceleration. This led me to believe that
derivatives somehow incorporate a temporal component with a 
spatial component, or to be simple a distance (meters) derives a 
velocity (meters per second) and then an acceleration (meters per 
second per second) Thus I see a derivative as incorporating a "per 
second" in some manner of calculation.

So now to my question. I had this thought that if you have an energy
component (joules), wouldn't the derivative of energy give you 
power (watts or joules per second)?

This seemed logical to me but since I am unfamiliar with the 
workings of calculus I could be wrong. 

Thank you for your help.

-Jeremy Entner

Date: 6/13/96 at 21:19:25
From: Doctor Luis
Subject: Re: derivatives/physics/time

As jfe2950@ritvax.rit.edu wrote to Dr. Math
On 2/2/96 at 21:5:21 (Eastern Time),

>I see a derivative as incorporating a "per 
>second" in some manner of calculation.

Indeed, the interpretation of the derivative depends on the physical
interpretation of the dependent and independent variables.

>If you have an energy component (joules),
>then wouldn't the derivative of energy give you power
>(watts or joules per second)?

 You are quite correct. Calculus (differential) deals with limiting
notions.  So, the derivative of position (r) gives you the 
"instantaneous" velocity dr/dt. If you remember power (P) being 
defined by 

   P = W / t

in differential notation you would write instead

  P = dW / dt      ("instantaneous" power)

Do not forget that calculus was developed by Sir Isaac Newton in 
order to deal with problems of motion, in particular those of the 
heavenly bodies, so it is logical that you reason the way you have.

-Doctor Luis,  The Math Forum