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What is an "Infinitesimal"?
{{Main|Derivative}}
{{Main|Derivative}}
Derivatives are defined as the instantaneous rate of change of differentiable functions. Derivatives themselves can be functions that give the slope of the instantaneous rate of change of a function at any differentiable point. The slope of a tangent line is another way of defining a derivative. A tangent line touches a graph at only one point (locally). So its slope can be interpreted as the instantaneous rate of change of that graph.
Derivatives are defined as the instantaneous rate of change of differentiable functions. Derivatives themselves can be functions that give the slope of the instantaneous rate of change of a function at any differentiable point. The slope of a tangent line is another way of defining a derivative. A tangent line touches a graph at only one point (locally). So its slope can be interpreted as the instantaneous rate of change of that graph.
==What is an "Infinitesimal"?==
The word "infinitesimal" is sometimes used in informal speech to mean "something extremely small", as in "The effect of the decline in soybean prices on the overall recession was infinitesimal." It's roughly a synonym for "inconsequential" in such a case.
But in calculus it has a somewhat clear, though not really precise meaning. It means some very small thing that is compared to another very small thing as they both decrease toward zero under some rule that connects them. It does '''not''' just mean "some very small number". It is '''not''' some kind of "opposite of infinity". Infinity is not a number either, but at least it behaves sort of like a number in that it can be compared to actual numbers. For any genuine number <math>X</math>, the question "Is <math>X < \infty</math>?" always has an answer of "yes"; that's the definition. But (ignoring negative numbers) if an infinitesimal were some kind of opposite of infinity in the sense of being infinitely small instead of infinitely large, the answer to the question "Is <math>X ></math> infinitesimal?" could only have an answer of "yes" in all cases if that infinitesimal were exactly zero. Which would make it a useless concept.
''An infinitesimal is a framework for doing differential calculus'', nothing more. It involves two very small quantities having a functional relationship to each other, such that they both decrease toward zero simultaneously. Their ratio is the derivative.
This concept was invented by Gottfried Leibniz with his "dx/dy" notation, and has been used ever since. Dx and dy are the infinitesimals. Isaac Newton's "fluxion" method of calculus did not involve this concept.
== External links ==
== External links ==