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Discussion: Research Area
Topic: Derivatives as quotients of infinitesimals

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Subject:   Newton Fluxions and Barron Infinitesimal Tangent Method
Author: Eric Goolish
Date: Jan 25 2004
The question you pose regarding a definition for dy/dx regarding infinitesimals
is very interesting.      
  As a quotient dy/dx, the derivative can be consider in terms of Newton's
Fluxions where he considered the tangent problem by the method of combining the
velocity components of a moving point in a suitable coordinate system.
Considering f(x,y)=0, Newton found that the motion of a particle is then the
composition of a horizontal motion with velocity vector having lenght 'X dot'
and a vertical motion with velocity vector having length 'Y dot.'  Then the
tangent to f(x,y)=0 at the point just becomes the sum of the horizontal and
vertical components.  Which yeilds 'Y dot'/'X dot.'  Since this is independent
of time, the derivative becomes dy/dx.

   I also found that Isaac Barrow's method of infinitesimal tangent methods
might be helpful in defining dy/dx.  He does not specifically mention the
quotient dy/dx, but he essentially employs the concept of the characteristic
triangle and neglects higher order infinitesimals to determine the tangent.
Essentially, he sets f(x+e, y+a) = f(x,y) = 0 and then ignores the higher order
infinitesimals.  He then solves the slope to be m = a/e. (which is essentially

   I hope this helps a little.  Post more questions if you have trouble
following my explanation.

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