Frank
You know that the term of dy/dx is absent of metrics unless limits are
applied.
If the object is to measure the plane at hand it obviously must
represent a homogeneous plane
when the limits applied. This was established long before I was born
so it was not I who made it up
Laugh away it is good for you.
Art

Your terminology is confusing: "The term of dy/dx". What does "term"
mean? How is it absent of metrics? I assume by "Metrics" you mean
a numeric value. dy/dx implies there is a function: y = f(x), for which
the derivative, f '(x) exists. The calculation of a derivative is trivial,
and assigning a numeric result simply involves substituting in
f ' (x) at x = a. I don't understand what you mean by applying limits
to a derivative. As long as the function is continuous, then the
derivative exists. Are you considering the "Newton Quotient"?
Why is the Newton Quotient relevant, when simple differentiation
methods will achieve the same answer.

What are you measuring in a plane? A plane is represented by a
linear equation in x, y, and z: such as:
a(x - xo)+b(y - yo)+c(z - zo) = 0. The coefficients a, b, and c
are a set of direction numbers of a normal to the plane.
Taking the derivative (dy/dx) of such a function implies a
"Partial" derivative, such that the "z" terms vanish, and you are
left with an equation of a line y = m*x+b, where the solution
is obviously "m". As for the homogeneity of a plane; you are introducing
a 4th dimension. What is the 4th variable? Subject to partial
differentiation
with respect to x; the 4th variable disapears anyway.

Did I get it right? I find the way you explain math is very difficult for
me to follow. Note: I am not laughing at you -- I assume you are
laughing at us who respond.

Frank