The discussion revolves around the interpretation and meaning of Leibniz notation in calculus, specifically focusing on the terms dy/dx, d2y/dx2, and the differentiation operator d/dx. Participants explore the implications of these notations in the context of derivatives and differential equations, as well as the conceptual understanding of differentials.
Participants exhibit a mix of agreement and disagreement regarding the interpretation of Leibniz notation and the operations involving differentials. While some points are clarified, there remains uncertainty and differing opinions on the legality of certain manipulations and the conceptual understanding of differentials.
Limitations include varying interpretations of the legality of operations involving differentials, the conceptual understanding of dy and dx, and the implications of treating these terms as quantities versus differentials.
Mark44 said:The d's stand for "differential". dy/dx is the derivative, not differential, of y with respect to x. The symbol d2y/dx2 represents the derivative (with respect to x) of the derivative with respect to x, or in other words, the 2nd derivative of y with respect to x. It could be written as
$$\frac{d}{dx}(\frac{dy}{dx})$$
As a notational shortcut, the above is often written as d2y/dx2.
d/dx is the differentiation operator, indicating that we're interested in taking the derivative (with respect to x) of whatever function is to the right of this operator.
Yes.BMW said:So the second derivative should really be d2y/(dx)2?
It means just what you would think it means - the derivative of y with respect to x.BMW said:If you have a differential equation with variables separated, such as dy/dx = 4x2/3y3, and you rearrange it to 3y3 dy = 4x2 dx, what does the dy/dx mean in this case,
I don't understand your question. The whole idea of separation of variables is to get all the expressions with y and dy on one side of an equation and all the expressions with x and dx on the other side - then integrate.BMW said:and can you even rearrange it like that or must you do this: ∫3y3 dy = ∫4x2 dx ?
Mark44 said:It means just what you would think it means - the derivative of y with respect to x.
The assumption here is that there is some differentiable function of x that is the solution to the differential equation. y represents that function.
I don't understand your question. The whole idea of separation of variables is to get all the expressions with y and dy on one side of an equation and all the expressions with x and dx on the other side - then integrate.
If it bothers you that you're integrating with respect to y on one side, but with respect to x on the other, you could think of the left side as being ∫3y3 dy/dx * dx, which simplifies to what you have above.
Possibly you've been told that dy/dx shouldn't be thought of as a fraction. Nevertheless, it's convenient to do so in many cases, such as in separating differential equations.
Mark44 said:It means just what you would think it means - the derivative of y with respect to x.
The assumption here is that there is some differentiable function of x that is the solution to the differential equation. y represents that function.
I don't understand your question. The whole idea of separation of variables is to get all the expressions with y and dy on one side of an equation and all the expressions with x and dx on the other side - then integrate.
If you mean you have the derivative, dy/dx= f(x), and you go to "dy= f(x)dx" by multiplying both sides by dx, yes, that, "multiplying by dx", is an "illegal operation" specifically because the derivative is NOT a fraction it is NOT "dy divided by dx". But, using the definition of the "anti-derivative" or "integral", we can go to \int dy= \int f(x)dx.iRaid said:It's probably this, when I took calculus in high school the teacher told us that this is an illegal operation.
BMW said:Sorry I should have explained it better. When you rearrange the equation, you get the x and dx on one side, and the y and dy on the other side. Do you have to make each side an integral? E.g. does it have to be ∫x dx = ∫y dy, or can you rearrange to x dx = y dy? If you can, what does the equation mean? What does multiplying x by dx do?
It seems weird to me that the dx and dy somehow magically fit into the integral (e.g. the dx which was part of a ratio now just tells you to integrate with respect to x). Does the dx on one side actually represent some quantity? Or is it more of a concept?
If y = f(x), then your first equation should be dy = f'(x)dx, and the second would be Δy = f'(x)Δxlavinia said:Rephrasing a little what has already been said.
As an equation among differentials, dy = f(x)dx is correct.
How ever dy and dx are not quantities. They are differentials of the functions y and x.
Differentials of functions are what get integrated, not quantities.
In Physics, the expression dy = f(x) dx is taken to mean that for very small Δy and Δx,
the equation Δy = f(x)Δx is approximately true and this approximation gets arbitrarily accurate for smaller and smaller Δx.
Δy/Δx approaches f'(x)lavinia said:In fact, Δy/Δx approaches f(x) arbitrarily closely as well.
dy/dx = f'(x)lavinia said:This approximation is expressed as the ratio of infinitesimals dy/dy =f(x)
lavinia said:which I would not be surprised actually meant something to Leibniz but nowsdays is taken merely as notation.