Is Derivative Notation Limited to Positive or Negative Values of dx?

[Calculuser]

dx>0 or dx<0 ??

I've just registered in this forum and I wanted to ask my question right away.I'm 18 and I love studying Calculus.While I was studying at Derivative part of it, I've realized something at Leibniz's Notation of Derivative ($\frac{dy}{dx}$).
It's equal to

lim $\frac{Δy}{Δx}$=$\frac{dy}{dx}$
Δx→0

My question is if we take the limit as Δx→0 (Δx→0$^{+}$ and Δx→0$^{-}$)
I think that's why dx must be both dx>0 and dx<0

Is it right??

Thanks..

 

[Stephen Tashi]

If you have questions like that, you are approaching calculus from an intuitive point of view. This may be Ok for your first exposure to it, but eventually you should also develop some skills for a precise approach to mathematics. (Pehaps you can start with a an introduction to Symbolic Logic.)

Although there are periodically various "bull session" type threads on forum about the "real" meaning of dy, dx or multiplication or whether irrational numbers exist etc,, if you are studying particular texts and they do not give any formal definition for "dy" and "dx" then these are simply symbols that have no independent interpretation. If you are asking about what Leibnitz himself thought about dy and dx, that is a question about the history of mathematical thought. That type of history is a very specialized field and what Leibnitz thought about these symbols is only a cultural background for the modern approach to calculus.

There is an approach to calculus called "nonstandard analysis" which defines an extended number system and may give formal definitions to dy and dx. Odds are, that the materials you are studying do not use that approach. If they don't, you can't sensibly interpret the definitions and assumptions from your text by using the definitions from "nonstandard analysis".

The intuitive approach to math tends to be "Platonic", i.e. it tends to think of mathematical objects and ideas as having a meaning that is independent of the formal definitions in books. The modern approach to mathematics is somewhat legalistic. The definitions of things are as they are written. Things aren't defined by people's ideas of "self evident" meanings or assumptions. So if something isn't defined, it's simply undefined. You can't take your private ideas about what it is and use it in a proof. (Of course, you always have the option to think about things using your own private notions.. It's just that such private notions aren't acceptable as arguments in formal proofs.)

 

[tiny-tim]

Welcome to PF!

Hi Calculuser! Welcome to PF!

My question is if we take the limit as Δx→0 (Δx→0$^{+}$ and Δx→0$^{-}$)
I think that's why dx must be both dx>0 and dx<0

Is it right??

Yes, that's correct.

In any metric space, when we say x -> a, we mean from any direction.

In this case, there are only two directions, but in ℝ² or ℝ³, there would be infinitely many directions.

 

[Calculuser]

Firstly, I should say that I'm glad to answer my new topic and thank you so much for this.

What's the name of the formal definition of "dx"?

Actually, I mean which books can help me to omit my all questions as I've mentioned in this topic.

I'm also proud of it's right by the way.

My question is if we take the limit as Δx→0 (Δx→0+ and Δx→0−)
I think that's why dx must be both dx>0 and dx<0

Is it right??

 

[3.1415926535]

The book Foundations of Infinitesimal Calculus by H. Jerome Keisler is a great introduction to Non standard analysis. I am not sure however if you have the mathematical maturity to read it...

 

[Stephen Tashi]

The book Foundations of Infinitesimal Calculus by H. Jerome Keisler is a great introduction to Non standard analysis. I am not sure however if you have the mathematical maturity to read it...

I'm curious whether it will say that dx > 0 and dx < 0.

 

[Calculuser]

The book Foundations of Infinitesimal Calculus by H. Jerome Keisler is a great introduction to Non standard analysis. I am not sure however if you have the mathematical maturity to read it...

I have this book and looked at it but I couldn't find answers to my questions.In fact, after I look at this book, questions filled up my mind.

 

[micromass]

There is no such thing as dx or dy or df. All it is, is a notation. We have the notation $\frac{dy}{dx}$ (which is a horrible notation), but this is NOT a fraction. The symbols dy and dx are NOT numbers. They are symbols, that's all.

Asking whether dx>0 or dx<0 is meaningless because dx is not a number.

So, while in normal calculus, dx has no meaning what-so-ever, we actually can give it meaning. But that is rather complicated. And even if we gave it a meaning, we would still not have that dx is a number (at least not in the conventional sense).

 

[Calculuser]

I'm really confused after all these messages.Somebody says so that it's a number but infinitesimal.

 

[micromass]

I'm really confused after all these messages.Somebody says so that it's a number but infinitesimal.

It's what I said in my last message.

In normal calculus, the notation dx has NO meaning what-so-ever. It is just a notation, nothing more.

However, people don't like this and what to give it meaning. This is why they introduce the concept of infinitesimal numbers. These are NOT ordinary real numbers. They are specially invented to give meaning to things like dx.

If you're confused then just remember that dx doesn't exist. The notation $\frac{dy}{dx}$ is not a fraction, but the notation for $\lim_{h\rightarrow 0}{\frac{y(x+h)-y(x)}{h}}$.

Compare it to this sitatuation:
When people are learning how to count, then they use things like 10-3. This notation is just a notation. It does not mean that there are actually things like "-3". People didn't like it, so they especially invented things like -3 to give a special meaning to 10-3. This does NOT imply that -3 is a natural number!

The difference is that numbers like "-3" are very succesful. They're used everywhere.
On the other hand, "infinitesimal numbers" which try to give meaning to dx are not so succesful. They're nice, but they're not commonly used in mathematics.

 

[Stephen Tashi]

I'm really confused after all these messages.Somebody says so that it's a number but infinitesimal.

I think you aren't reading the answers carefully.

When talking about the material in a traditional modern calculus text, no poster in the thread has said that dx is any kind of number.

tiny-tim answered "Yes, that's correct" but (I hope) he is referring to your statement that the derrivative is defined as a two sided limit. If he meant to say that "dx" is some sort of number in the traditional caclulus course, he is incorrect.

You yourself aren't using language precisely. The modern notation for the derivative of a real valued function f(x) is $\lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$. You prefer to write this as $lim_{\triangle x \rightarrow 0} \frac{\triangle y}{\triangle x}$. In these notations it is not correct to speak of h or $\triangle x$ as single numbers. In those definitions, they are variables. It makes sense to say a variable can take on both positive and negative values. It makes no sense to say that a single number can be both positive and negative at the same time. So you should stop speaking about $\triangle x$ as if it were a constant.

The use of the term "infinitesimals" is not part of the modern traditional approach to calculus. It was used in the time of Leibnitz, but that non-rigorous approach to calculus has been discarded. People still find it useful manipulate the dy and dx symbols as Leibnitz did. It helps you remember many of the rules of calculus. People doing physics still find it useful to reason as Leibnitz did about applications of calculus. However, from the point of view of proving things about calculus, "infinitesimals" are no longer used in the traditional approach.

You said you have the book by Jerome Keisler. You must realize that this book is a differnt approach to calculus than traditional texts use. Don't assume the definitions in that book are identical to the definitions in a traditional calculus course. A given area of mathematics can usually be approached in several different ways by authors who make different assumptions and definitions. You can't assume that different math books are consistent with each other. If your approach to studying a topic in mathematics is to read about it a variety of different books, you are going to have keep that in mind.

 

[Stephen Tashi]

...and, I see the reason for your confusion, if you are paying attention to all the posters in the thread on "Liebnitz notation". That is what I call a "bull session" type of thread. Some people in that thread are not trying to give precisely correct mathematical answers. They are stating how they think about things in their own mind. They (and you) are free to think about things however you wish, but unless you can formulate these thoughts into some sort of consistent mathematical system, they aren't valid mathematics.

 

[Calculuser]

Consequently, they don't have consensus on this subject, they use their approximations to imagine how it could be.

Thanks for your interests..

 

[3.1415926535]

In short, in the traditional approach to calculus(with ε-δ definition of limits, continuity etc.) "dx" has no meaning and is used only because Leibniz used it. In nonstandard analysis, this "dx" is taken to be any infinitestimal and there are both positive and negative infinitestimals.

 

[Calculuser]

Thank you that's what I've looked for.