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 email@example.com 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
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