Difference between notations

[3B1B]

Hi all, I am currently studying derivatives and have just begun,

I came across 3 different ways of denoting derivatives,

1. Lagranges notation: f'

2. Leibniz notation : dx/dy

3. Newton notation : y with a dot over it.


I would like to know the different uses and why we need three different ways of notation.

All help appreciated.

 

[R.M.]

1. I have always understood this to be a shortcut notation for a function's derivative. If I have some function f, then it's derivative is simply f'; it's second derivative is f", etc... This notation allows you to use the idea of a derivative of a function in a non-specific way. It is usually implied that f is a function of x, but that's not a rigid rule; f could be a function of t for example, so in that case I would write f'(t).

2. I think you mean dy/dx, but dx/dy is not necessarily incorrect. This notation is useful for being very specific about your function and derivative.
dy/dx: "the derivative of y with respect to the independent variable x"
dx/dy: "the derivative of x with respect to the independent variable y"
ds/dt: "the derivative of s with respect to the independent variable t" etc...

The first notation does not give you that specific information. This notation will also be helpful when you learn implicit differentiation, and also very helpful with differential equations. You can also think of this notation as [MATH]\Delta[/MATH]y/[MATH]\Delta[/MATH]x, with "dy" referring to a tiny change in y and "dx" referring to a tiny change in x.

3. I've never seen this notation outside a history of math course.

 

[HallsofIvy]

We don't need three different notations but it is simpler to write y' than \(\displaystyle \frac{dy}{dx]\). On the other hand it is simpler to write \(\displaystyle \frac{d^{10}y}{dx^{10}}\) than y with 10 's: y''''''''''!

The "dot" notation is primarily used in physics texts and then only for differentiation with respect to time.

 

[Dr.Peterson]

I came across 3 different ways of denoting derivatives,

1. Lagranges notation: f'

2. Leibniz notation : dx/dy

3. Newton notation : y with a dot over it.

I would like to know the different uses and why we need three different ways of notation.

Basically, these three different mathematicians just each invented a different notation, and that's why they all exist; each is useful in its own way, and that's why they all still exist! We could get along with only one, but each has its place.

Lagrange emphasizes that the derivative is a function "derived" from another. It is often the best way to formally state theorems. But it hides what variable you are differentiating with respect to (namely, whatever is the argument to f). It is also use in the form y', treating the variable y as a function.

Leibniz emphasizes the relationship of the derivative to a limit of a ratio of differences (the reason it's called "differentiation"). It is wonderful for keeping track of multiple variables, and makes the chain rule very memorable.

Newton's notation is used in physics, and is generally similar in usage to Lagrange, except that it doesn't focus on the name of the function, but of the variable; and it is only used with respect to time.

It happens that I wrote about these and other notations in my blog some time ago; I didn't explore them all equally deeply, but it will give you an overview: What Derivative Notations Mean.

 

[3B1B]

Thank you all for your replies, I believe I understand them now.

 

[JeffM]

I have seen Newton’s notation used in dynamics generally. Again it is is with respect to a function with an independent variable of time, but not necessarily restricted to physics. That is, it might involve biology or economics or any field where time is a controlling variable.

The main benefit of the notation is that keeps the focus on what is changing over time.