by danne89
 P: 181 A common definition I've read: A derivative of a arbitrary function is the change in one given instant. It's hard to think about it. I mean, movement, which is change of position cannot be defined for zero time. Has Zeros paradox something to do with this? If I don't misstaken, Zeros paradox is about the sum of infinity many parts can be finite. Very confusing...
 Sci Advisor HW Helper P: 9,396 Or it could be that your "common definition" is just an illustrative explanation and not acutally a rigorous definition at all.
 P: 181 Ahh. I see. But how does Zenos paradox relate?
HW Helper
PF Gold
P: 12,016

 Quote by danne89 Ahh. I see. But how does Zenos paradox relate?
It doesn't.

Note that in the "ordinary" interpretation of Zeno's paradox, that paradox is RESOLVED by noting the fact that, say, an infinite number of terms may add up to something finite.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 That's exactly WHY we need a rigorous definition for the derivative! In fact, that's exactly what led Newton to develop the calculus! He wanted to show that the gravitational force was dependent on the distance from the mass (and, of course, the acceleration due to that force). But the distance could (in theory anyway) measured at any instant while neither speed nor acceleration could, without calculus, be DEFINED at a given instant. The fact that "speed at a given instant" (and, therefore "acceleration at a given instant") could not even be defined was really Zeno's point and the calculus was a way to do that. Arildno, Zeno had several different "paradoxes". You, I think, are thinking of the one about "You can't cross this distance because you would first have to cross 1/2 of it, then 1/4 of it, then 1/8 of it, etc." danne89 was probably think of the "arrow" paradox: "At any given instant, the arrow is AT a specific point and therefore, not moving! Since it is not moving at any given instant, it can't be moving at all."
 P: 181 Nice. But as I've heard it, Newton's definiton wasn't rigorous.
 Sci Advisor HW Helper P: 11,948 Not in the terms of modern analysis.At that time,IT WAS RIGUROUS ENOUGH TO DELIVER THE CORRECT THEORETICAL RESULTS...Namely explaing the laws of Kepler. Daniel.
 P: 181 So todays rigurous definition may turne out to be non-exact tomorrow, when new physic demands avaible?