113

definition of derivative

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Posted (edited)

Derivative is defined as the limit of a finite difference

\[\frac{dy}{dx} = \lim_{h\to\0}\frac{f(x+h)- f(x)}{h}\]

where h is finite.

Is it possible to define the derivative by

\[\frac{dy}{dx} = \frac{f(x+dx)- f(x)}{dx}\]

where dx is not finite but infinitely small, infinitesimal ?

Edited by 113
I made a typo, I tried to correct it but it does not allow

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58 minutes ago, 113 said:

Is it possible to define the derivative by

 

dydx=f(x+dx)f(x)dx

 

where dx is not finite but infinitely small, infinitesimal ?

You mean \( \frac{df}{dx}(x)\). 

Maybe it is possible. But only after you explain the meaning of the expression on the right hand side. E.g. what exactly do you mean by saying that dx is "infinitely small, infinitesimal"? If there are several different dx that are "infinitely small, infinitesimal", then it would only make sense if you can prove that the value of the expression on the right hand side does not depend on which one of the possible values of dx that you apply. Obviously you have to explain what it means to do addition and division with objects that are "infinitesimal". You will have a lot of work to do before you can convince anyone that it makes sense. If you are willing to put in the work, then good luck. 

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Posted (edited)
21 minutes ago, taeto said:

You mean dfdx(x)

Yes.  Derivative is defined as the limit of a finite difference

\[\frac{dy}{dx} = \lim_{h\to\ 0}\frac{f(x+h)- f(x)}{h}\]

where h is finite.

Quote

Maybe it is possible. But only after you explain the meaning of the expression on the right hand side. E.g. what exactly do you mean by saying that dx is "infinitely small, infinitesimal"?

I mean that dx is not finite as opposed to h which is finite. If dx is not finite, it is infinite, infinitely small.

Edited by 113

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Posted (edited)
1 hour ago, 113 said:

Yes.  Derivative is defined as the limit of a finite difference

 

dydx=limh 0f(x+h)f(x)h

 

where h is finite.

     There is a "y" on the left hand side, but an "f" on the right hand side. The equation relates the derivative of a function y of x to the derivative of a function f of x. That is not what the definition of a derivative is supposed to look like. 

1 hour ago, 113 said:

I mean that dx is not finite as opposed to h which is finite. If dx is not finite, it is infinite, infinitely small.

What kind of thing is dx supposed to be? An integer, a rational number, a quaternion, what? What do you mean when you say "infinitely small"? Can you give any examples of mathematical objects that are "infinitely small"?

Edited by taeto

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13 minutes ago, taeto said:

     There is a "y" on the left hand side, but an "f" on the right hand side. The equation relates the derivative of a function y of x to the derivative of a function f of x. That is not what the definition of a derivative is supposed to look like. 

It makes no difference because y = f(x) so that dy/dx = df(x)/dx

13 minutes ago, taeto said:

What kind of thing is dx supposed to be? An integer, a rational number, a quaternion, what? What do you mean when you say "infinitely small"? Can you give any examples of mathematical objects that are "infinitely small"?

1/∞ is infinitely small. Is it 0 or non-zero?

The introduction of infinity brings a duality into the definition of an infinitesimal, meaning that we have to deal with objects that both zero and non-zero at the same time.

 

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Posted (edited)
1 hour ago, 113 said:

It makes no difference because y = f(x) so that dy/dx = df(x)/dx

You were hiding the information about \(y(x)\) being the same as \(f(x)\)?

1 hour ago, 113 said:

1/∞ is infinitely small. Is it 0 or non-zero?

I do not know the answer. What do you mean by \(\infty\)? And by \(1/\infty\)? Why do you think that \(1/\infty\) is infinitely small? What does it mean to you that something is infinitely small? You are saying things that have no place in usual mathematics. And do you know how to answer a question without introducing several more unknown quantities?

1 hour ago, 113 said:

The introduction of infinity brings a duality into the definition of an infinitesimal, meaning that we have to deal with objects that both zero and non-zero at the same time.

What do you mean when you speak of "infinity"? What is that exactly to you? I have absolutely no idea what you are going on about. Why would we have to deal with objects that have contradictory properties, it would seem an extraordinary stupid thing to do, no?

3 hours ago, 113 said:

I mean that dx is not finite as opposed to h which is finite. If dx is not finite, it is infinite, infinitely small.

Again now, look: the adjective "finite" in mathematics applies to sets. A set X is finite if there is a natural number n such that the "size" of X is n, which means that there exists a bijection from {1,2....,n} to X, so that you can count the elements of X from 1 to n. So correspondingly, the adjective "infinite" applies to a set for which there is no such n. Which kind of set is your dx for which the adjective "infinite" applies to it, and how can you prove this? And if so, how does this fact apply to the rest of the things that you are saying?

Edited by taeto

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Originally Newton tried to define the derivative as "dy divided by dx" where dy and dx are "infinitesmals" but was not able to give a rigorous definition of "infinitesimal".  The Bishop Berkeley famously satirized them as "ghosts of vanished quantities".  Later people like Cauchy redefined the derivative in terms of limits.  But recently Abraham Robinson and others resurrected "infinitesmals" by extending the real numbers to include both "infinite" and "infinitesmals" in "non-standard analysis":  https://en.wikipedia.org/wiki/Non-standard_analysis.

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Posted (edited)
1 hour ago, Country Boy said:

Originally Newton tried to define the derivative as "dy divided by dx" where dy and dx are "infinitesmals" but was not able to give a rigorous definition of "infinitesimal".  

I do not believe this is historically accurate. Leibniz used infinitesimals directly. Newton referred to the "ultimate ratio" of delta-y over delta-x as both become closer and closer to zero. Newton's idea has more in common with the modern viewpoint than Leibniz's. That's my understanding. There is historical debate around this topic.

Edited by wtf

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9 hours ago, taeto said:

You were hiding the information about y(x) being the same as f(x) ?

Who told you about \(\frac{dy}{dx}\) but hided the information about \(\frac{f(x+dx)- f(x)}{dx}\) ?

Was it me or was it all the math books? And why should they, the mathematicians,  have done so?

9 hours ago, taeto said:

I do not know the answer. What do you mean by ? And by 1/ ? Why do you think that 1/ is infinitely small? What does it mean to you that something is infinitely small? You are saying things that have no place in usual mathematics.

Why can't "usual mathematics" deal with something that is infinitely small?

9 hours ago, taeto said:

And do you know how to answer a question without introducing several more unknown quantities?

What is the question? Is the question "is 1/∞ either 0 or non-zero?" ?  The question does not have an unambiguous answer either 0 or non-zero, because 1/∞  is both.

9 hours ago, taeto said:

What do you mean when you speak of "infinity"? What is that exactly to you? I have absolutely no idea what you are going on about. Why would we have to deal with objects that have contradictory properties, it would seem an extraordinary stupid thing to do, no?

If that is a stupid thing to do, then why don't you answer the question: "is 1/∞ either 0 or non-zero?" ?

9 hours ago, taeto said:

Again now, look: the adjective "finite" in mathematics applies to sets.

Yes. But I am not talking about sets,  because maybe it is not necessary right now. I am talking about finite difference f(x+h) - f(x) where h is finite, as opposed to f(x + dx) - f(x) where dx is not finite. Adjective finite refers to distance in this case, finite distance.

9 hours ago, taeto said:

 

A set X is finite if there is a natural number n such that the "size" of X is n, which means that there exists a bijection from {1,2....,n} to X, so that you can count the elements of X from 1 to n. So correspondingly, the adjective "infinite" applies to a set for which there is no such n. Which kind of set is your dx for which the adjective "infinite" applies to it, and how can you prove this? And if so, how does this fact apply to the rest of the things that you are saying?

You can count the elements of X only if you know n. But because you did not tell what is n, I can't tell how many elements are there in X. So it may look as though dx

is a set for which you don't tell what is n.

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2 hours ago, 113 said:

Who told you about dydx but hided the information about f(x+dx)f(x)dx ?

Was it me or was it all the math books? And why should they, the mathematicians,  have done so?

If your math book suggests to compute the derivative of \(y\) as the limit of the difference quotient of \(f\), without saying first that \(y=f(x)\), then it makes sense for you to throw it in the recycling bin.

2 hours ago, 113 said:

Why can't "usual mathematics" deal with something that is infinitely small?

Maybe it can. But this is analysis, a particular branch of math that deals with functions of real and complex numbers. If you ask whether it makes sense to have a real number \(d\) with the property \(0 < d \leq x\) for all real numbers \(x\), then the answer is that such a \(d\) cannot exist, because, e.g., \(d/2\) has \(0 < d/2 < d\) in \(\mathbb{R}\) contradicts that \(d\) has the required property. 

If this is not the property that you would want an "infinitely small" number to have, then what is the property that you are thinking of? Will you evade to answer that question? 

2 hours ago, 113 said:

What is the question? Is the question "is 1/∞ either 0 or non-zero?" ?  The question does not have an unambiguous answer either 0 or non-zero, because 1/∞  is both.

Just writing up some string of symbols like \(1/\infty\) doesn't always point to something that makes any sense. I ask again, what does it mean to you? Apparently you can make sense of it, since you keep going on about it. 

2 hours ago, 113 said:

If that is a stupid thing to do, then why don't you answer the question: "is 1/∞ either 0 or non-zero?" ?

Why don't you answer the question: "what does it mean?"? 

2 hours ago, 113 said:

Yes. But I am not talking about sets,  because maybe it is not necessary right now. I am talking about finite difference f(x+h) - f(x) where h is finite, as opposed to f(x + dx) - f(x) where dx is not finite. Adjective finite refers to distance in this case, finite distance.

Now it looks like you use "finite" to mean "non-zero". In the context of analysis, distance is given by Euclidean metric, and then the only distance that is not "finite" is identically 0. You end up with \(f(x+dx)-f(x) = 0\) always, independently of \(f\) and \(x\). 

2 hours ago, 113 said:

So it may look as though dx is a set for which you don't tell what is n.

The most positive I can say would be something like just forget about the actual meaning of \(dx\). In ordinary usage, the functions \(x\) and \(dx\) belong to different species, they do not allow to be composed together by the binary operation of addition. It is something like trying to add a scalar to a 2-dimensional vector. 

The most you could do is to use your expression as a notational shorthand, which substitutes \(dx\) for \(h\) and removes the needs to write the \(\lim\) symbol. Maybe you can think of any advantages in doing so. Something like how the Leibniz notation allows to write the chain rule in the form \(\frac{dz}{dx} = \frac{dz}{dy}\frac{dy}{dx}\).

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113,

It would help to know if you are starting calculus at high school or restudying the subject more deeply at university?

It may seem silly and obvious to say that when you start something you have to start somewhere.

The way calculus was developed and the way we learn it is not the way it is formally collected together and written down in textbooks.

So when you start (or restart study) there is much they don't tell you at first.

This is not due to any malevolent intent, it is simply to make the presentation understandable.

One of the reasons that differentiation is presented in the form it now appears is because there are many types of differentiation.

You are studying the functions of a single real variable.

The modern form can be extended to differention of Vectors (and tensors if you have heard of them) for instance.

The idea is that we make a (small) change to the variable concerned and compare the result in the function of that variable to the starting point.

With a single real variable there is only one sort of change that can be made, so implementation of the modern formula looks a bit over the top.

But coming on to more advanced forms of differentiation,  the 'derivative' is not even a single function but a matrix (called a Jacobian) or other multidimensional entity.

 

On a historical note.

Newton was basically a Physicist, who needed to invent most of the Maths he needed.

He came to calculus of functions from the calculus (there are several calculi) of finite differences, which he invented. There are many formulae in this calculus which bear his name.
He was doing this because he was interested in producing (interpolating) tables of values he needed and he was one of the pioneers of this subject.
With the completion of most tables and the advent of computers, Finite Difference methods are not taught so commonly today.

Leibnitz was primarily a mathematician, studying and developing the 'analysis' of his day.
So he came to calculus directly from the theory of functions as it then existed as graphs drawn on paper.

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On 10/7/2018 at 12:37 PM, studiot said:

The idea is that we make a (small) change to the variable concerned and compare the result in the function of that variable to the starting point.

With a single real variable there is only one sort of change that can be made, so implementation of the modern formula looks a bit over the top.

What if we make a change equal to dx to the variable x? Doesn't the function f(x) change equal to f(x + dx) ?

 

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29 minutes ago, 113 said:

What if we make a change equal to dx to the variable x? Doesn't the function f(x) change equal to f(x + dx) ?

Could you demonstrate how that works if \(f(x)\) is equal to \(x^2\) for every real number \(x\): how would you calculate \(f'(0)\) using your formula?

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1 hour ago, 113 said:

What if we make a change equal to dx to the variable x? Doesn't the function f(x) change equal to f(x + dx) ?

 

Yes inded it does.

So the next step is to find an expression for  f(x+dx).

Using taeto's example we have

f(x+dx) is (x+dx)2

or

x2 + 2xdx + (dx)2.

Now you need to take a very important step in this method or route to the derived function.

If dx is a very small quantity then (dx)2 is insignificant and may be ignored so we have

the top line of your fraction is

f(x) - f(x+dx) = x2 - (x2 + 2xdx) = 2xdx.

So

[math]\frac{{dy}}{{dx}} = \frac{{2xdx}}{{dx}} = 2x[/math]

 

Now this is an example of 'starting somewhere' and glossing over certain parts of the derivation.

It justifies the use of the notation y = f(x) as the above can get very clumsy.

It is normal to use   [math]\delta x[/math]   not dx and take a limit as [math]\delta x[/math]  approaches zero, before limits of ratios have been properly studied and quickly say

as [math]\delta x[/math] and [math]\delta y[/math]  tend to zero


[math]\mathop {\lim }\limits_{\delta x \to 0} \frac{{\delta y}}{{\delta x}} = \frac{{dy}}{{dx}}[/math]

 

But this has gone out of fashion.

 

 

 

 

 

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7 hours ago, studiot said:

If dx is a very small quantity then (dx)2 is insignificant and may be ignored

That is the physicist's approach.

From a mathematical point of view, the quantity \((dx)^2\) does depend on \(dx\), and `ignorable' is not a valid predicate. So from that point, the derivative becomes a function of the variable \(dx\).

I suspect that the OP wants to consider the possibility that \(dx\) is not a real number. That leaves them with the unthankful burden of explaining the meaning of evaluating \(f(x+dx)\) for a function \(f\), which is only defined on real numbers, in a point \(x+dx\) which is not a real number. 

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31 minutes ago, taeto said:

That is the physicist's approach.

From a mathematical point of view, the quantity (dx)2 does depend on dx , and `ignorable' is not a valid predicate. So from that point, the derivative becomes a function of the variable dx .

I suspect that the OP wants to consider the possibility that dx is not a real number. That leaves them with the unthankful burden of explaining the meaning of evaluating f(x+dx) for a function f , which is only defined on real numbers, in a point x+dx which is not a real number. 

Well I have to differ here on several counts.

113 is not using terminology exactly for instance

 

On 06/10/2018 at 5:32 PM, 113 said:

Derivative is defined as the limit of a finite difference

Clearly misusing the term finite difference, which is a term concerning real numbers.

I think 113 meant 'non zero difference' here.

 

The approach I described was in use in grammar schools in England from at least 1880 to the mid twentieth century, for the subject of Pure Mathematics, my introduction to calculus was done that way in the mid 1960s. As I said, university analysis revisited the whole issue much more rigourosly.

Throughout the twentieth century there was much discussion as to presentation and the use of infinitesimals, various delta x s,  a small quantity h, various greek letters and so on.

As I said fashions change.

 

The bottom line was, and remains, that when pupils start this subject there are a great deal if disparate subjects within the discipline to get their heads around very quickly.

So corners are always initially cut somewhere.

We had become proficient in differentiation within the first term ie between September and Christmas in my 1960s A level.
To do that we had a substantial amount of new more advanced algebra (sequences, series , limits ) and new trigonometry to get through at the same time.

Integration came after Christmas.

 

:)

 

 

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18 minutes ago, studiot said:

Well I have to differ here on several counts.

113 is not using terminology exactly for instance

If anything I said so far can be construed to implicate that I think that 113 uses terminology which is in any way consistent or meaningful, then I apologize.

What are other counts by which you differ?

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29 minutes ago, taeto said:

If anything I said so far can be construed to implicate that I think that 113 uses terminology which is in any way consistent or meaningful, then I apologize.

What are other counts by which you differ?

I can see you are trying to point 113 towards more advanced and more coherent thinking I just don't think he has enough mathematics to take all that in.

My point of departure is simply that until 113 tells us where he is coming from any attempt to help is only guesswork, so I have tried to be open about what I am guessing and the context in which I am pitching an explanation.

I am not a teacher, but judging by the many textbooks on the subject I have seen, 113's questions are very common and nothing to be ashamed of.

Edited by studiot

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2 hours ago, studiot said:

I can see you are trying to point 113 towards more advanced and more coherent thinking I just don't think he has enough mathematics to take all that in.

My point of departure is simply that until 113 tells us where he is coming from any attempt to help is only guesswork, so I have tried to be open about what I am guessing and the context in which I am pitching an explanation.

I am not a teacher, but judging by the many textbooks on the subject I have seen, 113's questions are very common and nothing to be ashamed of.

It would be interesting to observe how he reacts to your own nice exposition re differentiating \( x \mapsto x^2\) at the point \(x=0\). But you are presenting a very classical approach, and I think that he is not aligned with that. I am more concerned with how he will explain the ways in which he diverges from the common understanding. Your explanation he will probably just blankly contradict. I would like to see his own explanation. 

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13 minutes ago, taeto said:

It would be interesting to observe how he reacts to your own nice exposition re differentiating xx2 at the point x=0 . But you are presenting a very classical approach, and I think that he is not aligned with that. I am more concerned with how he will explain the ways in which he diverges from the common understanding. Your explanation he will probably just blankly contradict. I would like to see his own explanation. 

It's not the way I usually try to explain it, I was trying to follow what I admit I am guessing is 113's approach by answering direct questions and correcting some notational errors.

 

There is an old saying

"Never ask an Irishman for directions"  

Because they will always say, "If I was going to XXX, I wouldn't start from here"

 

My usual explanation is to start from the idea of an expression giving specific values  (eg y = x2 say when x = 0)

Then to move on to the idea of a function as a whole entity in itself so y = x2 for all x

and then introduce the idea of the derived function - which is also a function in its own right.

 

But my ever hopeful bottom line is that I don't (want to) see 113 as one of that merry band of folks dedicated to disproving, Cantor, Leibnitz Reiman et al, come what may.

Edited by studiot

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4 hours ago, studiot said:

Well I have to differ here on several counts.

113 is not using terminology exactly for instance

How exact is your

Quote

If dx is a very small quantity then (dx)2 is insignificant and may be ignored so we have

Tell exactly how small quantity must dx be so that (dx)^2 can be ignored.

 

Edited by 113
correcting a typo

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1 hour ago, 113 said:

How exact is your

It helps to finish your sentences.

It also helps to answer the questions of others if you want your own answered.

1 hour ago, 113 said:

Tell exactly how small quantity must dx be so that (dx)^2 can be ignored.

Since I still don't know where you are coming from;

I suggest you read the standard Oxford University text at this level for half a century.

An introduction to the Infinistesimal Calculus

by

G W Caunt

Oxford University press

Chapter 3

Differentiation of simple algebraic functions

In general and section 24 pages 64 to 67 in particular

Differentials and Orders of small quantities.

 

 

 

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4 hours ago, studiot said:

I suggest you read the standard Oxford University text at this level for half a century

That'll keep him busy.

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8 hours ago, studiot said:

It helps to finish your sentences.

Alright:

How exact is your " If dx is a very small quantity then (dx)2 is insignificant and may be ignored so we have" ?

I thought that you would understand that my sentence continued below.

Quote

It also helps to answer the questions of others if you want your own answered.

Since I still don't know where you are coming from;

I suggest you read the standard Oxford University text at this level for half a century.

An introduction to the Infinistesimal Calculus

so does it tell exactly how small dx is?

Edited by 113

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On 10/7/2018 at 11:37 AM, studiot said:

On a historical note.

Newton was basically a Physicist, who needed to invent most of the Maths he needed.

<NitPickerMode>

Newton was basically an alchemist and theologian.

On 10/7/2018 at 11:37 AM, studiot said:

Leibnitz was primarily a mathematician, studying and developing the 'analysis' of his day.

Debatable, one should know how much time and effort he placed in the different disciplines (which I don't know).

A difference with Newton is that Leibniz stood fully in the academic disciplines of his day, where Newton, e.g. in his theology would have been seen as a heretic (and therefore Newton decided not to publish his theological writings). In his theological studies Newton e.g. denied the Trinity, based on textual criticism. Modern New Testament scholars agree with his conclusions.

</NitPickerMode>

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