Leibniz Notation Explained: dy/dx, d2y/dx2, d/dx

[BMW]

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]

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.

 

[BMW]

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.

So the second derivative should really be d²y/(dx)²?

 

[BMW]

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, and can you even rearrange it like that or must you do this: ∫3y³ dy = ∫4x² dx ?

 

[Mark44]

So the second derivative should really be d²y/(dx)²?

Yes.

 

[Mark44]

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,

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.

and can you even rearrange it like that or must you do this: ∫3y³ dy = ∫4x² dx ?

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.

 

[iRaid]

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.

It's probably this, when I took calculus in high school the teacher told us that this is an illegal operation.

 

[BMW]

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.

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?

 

[HallsofIvy]

It's probably this, when I took calculus in high school the teacher told us that this is an illegal operation.

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 $\int dy= \int f(x)dx$.

It is because of the fact that, while the derivative is NOT a fraction, it can always be treated like one, that we define the "differentials", dy and dx separately so that, once we have that definition, we can think of "dy/dx" as a fraction. It is a "mixed notation" that can be confusing, but useful.

 

[lavinia]

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?

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. In fact, Δy/Δx approaches f(x) arbitrarily closely as well. This approximation is expressed as the ratio of infinitesimals dy/dy =f(x) which I would not be surprised actually meant something to Leibniz but nowsdays is taken merely as notation.

 

[Mark44]

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.

If y = f(x), then your first equation should be dy = f'(x)dx, and the second would be Δy = f'(x)Δx

In fact, Δy/Δx approaches f(x) arbitrarily closely as well.

Δy/Δx approaches f'(x)

This approximation is expressed as the ratio of infinitesimals dy/dy =f(x)

dy/dx = f'(x)

which I would not be surprised actually meant something to Leibniz but nowsdays is taken merely as notation.