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Topic: [ap-calculus] Implicit Differentiation notation
Replies: 2   Last Post: Feb 9, 2012 7:39 PM

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 BCrombie@AOL.COM Posts: 108 Registered: 12/8/04
Re: [ap-calculus] Implicit Differentiation notation
Posted: Feb 9, 2012 7:39 PM

This is a case where a careful examination of context and purpose makes all
the difference in the world.

In the process of defining the derivative as the limit of the slopes of
secant lines, procedurally, the expression dy/dx can not be regarded as a
quotient. True enough. But this is not the context, conceptually or
procedurally, for the stated problem.

The approach of the student (though he certainly doesn't realize this) is
an application of Analytic Geometry, plain and simple. Given the curve,
2xy+4y^2=5, the equation 2(dx)y+2x(dy)+8y(dy)=0 is the equation of the tangent
line to the curve.The displacements, dx and dy, are horizontal and vertical
displacements that leave the tangent line at a point and return to the
tangent line at a point as surely as delta x and delta y do for the graph of
y=mx+b. If you actually want to see the equation of the tangent line use
the substitution dx=x-x0 and dy=y-y0. There are no infinitesimals running
about in this argument. It is strict Analytic Geometry.

There is a reasonable reason why "the Leibnitz notation is robust enough
that all algebraically correct notational abuses for differentiable functions
of one variable will actually give the correct result." The reason is
'cause it's just the Analytic Geometry of finite variables x and y. The usual
circumlocution is necessary in correctly defining the limit of the slope of
secant lines. It is both unnecessary and a source of confusion to students
when uncritically applied outside of its role as definition. Let's not
confuse the kids any more then is absolutely necessary. In fact, it would
probably help them immensely to see that every thing they learned in Algebra I
about straight lines and linear functions is still true, and at times even
useful, in the Calculus.

Bill

PS: A note about implicit differentiation. For an algebraic function, as in
this example, we don't even have to differentiate to find the equation of
the tangent line and consequently its slope. That may be interesting
because the implicit differentiation misses vertical tangent lines. But that is a
lot more Analytic Geometry.

In a message dated 2/9/2012 6:08:32 P.M. Eastern Standard Time,
jeffreyrcagle@gmail.com writes:

He probably did see something about separable differential equations, and
then transferred it over to implicit differentiation. Or else, the author
of his source did.

Here's the thing: dy/dx is not actually the quotient of two quantities
"dy" and "dx". However, the Leibnitz notation is robust enough that all
algebraically correct notational abuses *for differentiable functions* *of one
variable* will actually give the correct results.

So this would "work": 2(dx)y + 2x(dy) + 8y(dy) = 0

2y(dx/dy) + 2x + 8y = 0
dx/dy = (-2x - 8y)/(2y)

successfully finds dx/dy, the derivative of the inverse of y.

And even,

2(dx)y + 2x(dy) + 8y(dy) = 0
2(dx/dt)y + 2x(dy/dt) + 8y(dy/dt) = 0

would solve a related-rate problem.

So, you ask, if the notation works then why is the differential method
incorrect?

Only because it conveys the wrong picture, that dy/dx is the quotient of
two little infinitesimal quantities. AND, if that error propagates upward,
it causes confusion with partial derivatives. As in, @f/@x does not equal
@x/@f.

But many physicists cheerfully ignore this point and treat dy and dx as
separate quantities anyways.

Jeff Cagle

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