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.
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, firstname.lastname@example.org 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.
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.