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