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BMW
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I never really understood leibniz notation. I know that dy/dx means differential of y with respect to x, but what do the 'd's mean? How come the second-order differential is d2y/dx2? What does that mean? And what does d/dx mean?
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 [itex]\int dy= \int f(x)dx[/itex].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.
Leibniz notation is a mathematical notation used to represent derivatives and differentials. It was developed by German mathematician Gottfried Wilhelm Leibniz in the 17th century.
"dy/dx" represents the first derivative of a function y with respect to the independent variable x. It is also known as the derivative of y with respect to x.
"dy/dx" represents the first derivative, while "d2y/dx2" represents the second derivative of a function y with respect to the independent variable x. In other words, "d2y/dx2" is the derivative of "dy/dx".
Leibniz notation is used to represent derivatives and differentials in calculus. It allows for a more intuitive and flexible approach to solving problems involving rates of change and optimization.
No, there are other notations for derivatives such as Newton's notation and Lagrange's notation. However, Leibniz notation is the most commonly used notation in calculus due to its simplicity and flexibility.