Is Standard Calculus Notation Wrong?

_{Jonathan Bartlett

April 8, 2019

Intelligent Design, Mathematics}

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We usually think of basic mathematics such as introductory calculus to be fairly solid. However, recent research by UD authors shows that calculus notation needs a revision.

Many people complain about ID by saying, essentially, “ID can’t be true because all of biology depends on evolution.” This is obviously a gross overstatement (biology as a field was just fine even before Darwin), but I understand the sentiment. Evolution is a basic part of biology (taught in intro biology), and therefore it would be surprising to biologists to find that fundamental pieces of it were wrong.

However, the fact is that oftentimes fundamental aspects of various fields are wrong. Surprisingly, this sometimes has little impact on the field itself. If premise A is faulty and leads to faulty conclusions, oftentimes workaround B can be invoked to get around A’s problems. Thus, A can work as long as B is there to get around its problems.

Anyway, I wanted to share my own experience of this with calculus. Some of you know that I published a Calculus book last year. My goal in this was mostly to counter-act the dry, boring, and difficult-to-understand textbooks that dominate the field. However, when it came to the second derivative, I realized that not only is the notation unintuitive, there is literally no explanation for it in any textbook I could find.

For those who don’t know, the notation for the first derivative is . The first derivative is the ratio of the change in y (dy) compared to the change in x (dx). The notation for the second derivative is . However, there is not a cogent explanation for this notation. I looked through 20 (no kidding!) textbooks to find an explanation for why the notation was the way that it was.

Additionally, I found out that the notation itself is problematic. Although it is written as a fraction, the numerator and denominator cannot be separated without causing math errors. This problem is somewhat more widely known, and has a workaround for it, known as Faa di Bruno’s formula.

My goal was to present a reason for the notation to my readers/students, so that they could more intuitively grasp the purpose of the notation. So, I decided that since no one else was providing an explanation, I would try to derive the notation myself.

Well, when I tried to derive it directly, it turns out that the notation is simply wrong (footnote – many mathematicians don’t like me using the terminology of “wrong”, but, I would argue that a fraction that can’t be treated like a fraction *is* wrong, especially when there is an alternative that does work like a fraction). Most people forget that is, in fact, a quotient. Therefore, the proper rule to apply to this is the quotient rule (a first-year calculus rule). When you do this to the actual first derivative notation, the notation for the second derivative (the derivative of the derivative) is actually . This notation can be fully treated as a fraction, and requires no secondary formulas to work with.

What does this have to do with Intelligent Design? Not much directly. However, it does show that, in any discipline, there is the possibility that asking good questions about basic fundamentals may lead to the revising of some of even the most basic aspects of the field. This is precisely what philosophy does, and I recommend the broader application of philosophy to science. Second, it shows that even newbies can make a contribution. In fact, I found this out precisely because I *was* a newbie. Finally, in a more esoteric fashion (but more directly applicable to ID), the forcing of everything into materialistic boxes limits the progress of all fields. The reason why this was not noticed before, I believe, is because, since the 1800s, mathematicians have not wanted to believe that infinitesimals are valid entities. Therefore, they were not concerned when the second derivative did not operate as a fraction – it didn’t need to, because it indeed wasn’t a fraction. Infinities and infinitesimals are the non-materialistic aspects of mathematics, just as teleology, purpose, and desire are the non-materialistic aspects of biology.

Anyway, for those who want to read the paper, it is available here:

Bartlett, Jonathan and Asatur Khurshudyan. 2019. Extending the Algebraic Manipulability of Differentials. Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis 26(3):217-230.

I would love any comments, questions, or feedback.

Comments

johnnyb, Referring to equation (6) on page 223 of the paper linked in the OP, you derive a formula for the third derivative. Did you explore even higher derivatives, and perhaps search for a compact way to express these derivatives? For example, I see 1s and 3s, which are the binomial coefficients 3 choose k for k = 0 up to 3. Perhaps there are some interesting connections between your formulas and the binomial theorem or other well-known identities. Maybe you could view all this as a system somewhat like the Weyl algebra, which is a ring of differential operators that arises in QM.daveS_{April 10, 2019
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JB, thanks, would appreciate. KFkairosfocus_{April 9, 2019
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H & JB, I'd start with something more basic, a tap and a cylindrical glass, then run water into it. The idea of a flow, a rate, of change and its accumulation as something concrete. With time as independent variable (onward, x(t) and y(t) thus trajectories). Then, a diagram, a leaky tank with inflow and outflow. Then, a gaussian pulse of water in, outflow locked off. Graphs -- yes graphs -- on parallel time axes: change and accumulation with causal correlations . . . and yes those will speak onwards in economics, marketing, sociotechnical systems, physics, biology, generalised growth, even statistics and more. Yes, ye olde water closet is relevant, and the effect of an outflow pulse counts too. Besides a bell curve or the like is often what a real world acceleration looks like -- corner and jump discontinuity headaches. It's not just Mathematics that gets a say here. Connexions count. From this what the slope of a curve is and what the area under it is can be analysed and will make sense. I have a sneaking feeling this is part of why I see such structures and quantities as part of the fabric of reality, I have been seeing connexions and cases. We can then go to the more conventional stuff. KFkairosfocus_{April 9, 2019
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It actually is quite important for those students who plan to major in mathematics.
Just to note, we already had this discussion, and I agreed with you it was important for the future (comment 17). The point of the present post (comment 25) was that, for a student who hasn't actually done any calculus yet, introducing such at this point is pretty meaningless, because they have no context for why it is important. Likewise, teaching calculus using infinitesimals, I don't actually teach a formal theory of infinitesimals until the last third of the semester. I mention the idea in passing when we are first getting started, with the promise that, if they want to do this more rigorously, we will return to the topic at the end of the course.johnnyb_{April 9, 2019
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JB writes, " First of all, in every single Calculus course I’ve ever seen, they are taught limits first. This includes high-school students taking Calculus." I don't think this is true, and it is certainly not how I taught calculus at the high school level. I agree with kf and you that we should start with intuitive ideas and simple equations and build understanding of the fundamental meanings of calculus (rate of change and accumulated amounts). We should build solid mechanics of the basic operations, and then add more rigorous theory later, at appropriate times. Also, even at the college level the university that I am familiar with has two calculus tracks: a rigorous one for math and physical science majors, and a more informal one for people like business and social studies majors. I remember once helping a former high school student come to me about being lost in college calculus I on a certain point. I was able to put it back into the context of the meanings I had taught her the previous year, and then she was clear and ready to tackle the mechanics because the overall purpose made sense. Eventually I completely abandoned the textbook we had and used all my own materials. One aspect of my situation was that many of my students were not actually going to take calculus in college, but the calculus they took with me strengthened their algebra skills and knowledge of functions considerably so that they were very well prepared for college algebra. I also used lots and lots of real-world examples for them to understand how the ideas that calculus were around them all the time. Also, I too often introduced limits right at the end of the year, including at least the idea of epsilon/delta proofs, because I knew that might be the one thing they would run into at the start of their college calc course that they hadn't seen. By that time they had become quite comfortable with thinking informally about infinitesimals and limits that they didn't have a problem with understanding the formalities.hazel_{April 9, 2019
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They get this weird mathematical formalism that has almost no point whatsoever, except to prove something they haven’t gotten to yet.
:o It actually is quite important for those students who plan to major in mathematics.daveS_{April 9, 2019
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Kairosfocus - A few things. First of all, in every single Calculus course I've ever seen, they are taught limits first. This includes high-school students taking Calculus. This focuses on epsilon-delta proofs. This is the first thing that Calculus students hit. It's a travesty. They get this weird mathematical formalism that has almost no point whatsoever, except to prove something they haven't gotten to yet. I move limits to the very end of my calculus course (I'll send you a copy of my book, Calculus from the Ground Up, if you are interested). I think your physical examples of rate vs accumulation is an excellent one. I'm an equations guy, so I usually start with the equation "y = 2", and then help the students build up an equation for the area under that line. That gives "y = 2x". I then help students build up an equation for the area under that line, which is "y = x^2". I then start plotting slopes under "y = x^2", and show that it pretty much gets close to "y = 2x". Then I ask what the slope of "y = 2x" is, which they should know from the equation of a line. Viola. It turns out they already know some calculus :) I'm pretty lazy, though, and I do almost all of my teaching on a whiteboard. No field trips for my students ;)johnnyb_{April 9, 2019
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JB, what level (and what age) is the Calculus usually first taught in the US system? It seems to me the epsilon-delta, limits approach with sequence, series and limits in required background, comes later after a more intuitive view of such concepts? Yes, that feeds forward to Analysis and a lot of other things, but itself does not seem to me to be conceptually, educationally foundational. I for example used to do a vest pocket demo on water pouring into a cylindrical glass at a variable rate in a gaussian pulse then plotting RATE vs ACCUMULATION of flow, and showing the fundamental theorem intuitively as two inverse operations. The graphs -- with time as natural independent variable, then allowed getting rate per chord to tangent and area under through accumulation of strips. Of course a Gaussian pulse then gives us a sigmoid and contrasts a maximum and an inflexion, with connexions. Move to a leaky tank and a negative pulse can be considered, put both together with a controller and we are at control systems . . . my first practical assignment was take the lid off your toilet tank (and if you don't learn a bit about it, you haven't really done practical control systems -- about the commonest mechanical, process control system). It is also a physically real event. The use of limits then comes out. Such a case study helps to set up a ring of key ideas that can then be drawn on. For example time as underlying variable allows pondering trajectories, growth, saturation etc, a very dynamic view relevant in many fields of thought. The sequences, series, limits, epsilon delta approach then comes out. Nonstandard analysis feeds in on infinitesimals having some substance as dt where [dt]^2 --> 0, which can be seen on how say 10^-6 squared is 10^-12 and this squared is 10^-24, etc, and how the span gets much wider as we go on down. Thoughts? KFkairosfocus_{April 9, 2019
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PaV - Regarding footnote 5, I'm not sure your point or question. That footnote basically reiterates the point of agreement between me and DaveS expressed in the last paragraph of comment #15 above.johnnyb_{April 9, 2019
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PaV - Great questions. Most people don't realize that Liebnizian calculus actually didn't have derivatives. It just had differentials. So, in my own writing, I separate out "differentiating" from "taking the derivative". For me, the derivative is a ratio of differentials. dy/dx is the ratio of the change in y to the change in x in the smallest units. However, to *get* dy and dx, you differentiate. So, if you had "sin(x) = y^2", you can differentiate this into "cos(x) dx = 2y dy". You could then solve for "dy/dx" algebraically. "dy/dx = cos(x) / 2y". This is a lot more straightforward for students, as it unifies explicit, implicit, and multivariable derivatives. So, the general form of d(u/v) is found using the quotient rule: (v du - u dv) / v^2. So, if we do d(dy/dx) we get "(dx d(dy) - dy d(dx))/dx^2". Since we are differentiating and *then* dividing by "dx" (because we are finding the change in the derivative vs. the change in x), it becomes "(dx d(dy) - dy d(dx))/dx^3". If you take d(dy) = d^2y and d(dx) = d^2x, and then simplify, this becomes "d^2y/dx^2 - (dy/dx)(d^2x/dx^2)", which is my formula. As for Faa di Bruno's formula, there is an example in the paper in Section 4 (see specifically equation 4) which shows how this form allows transformations without the formula. For instance if we started out with "y = x^3" and "x = t^2", we could combine their derivatives algebraically without usage of Faa di Bruno's formula. It's all doing basic algebra and cancelling.johnnyb_{April 9, 2019
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JohnnyB: I just saw your footnote #5. So, a distinction is being made. When is this distinction necessary and used by mathematicians?PaV_{April 9, 2019
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PS to my #19 To sum up, to those students for whom this approach works, more power to them.daveS_{April 9, 2019
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johnnyb, It does take time to teach these "non-standard" methods, however, which could crowd out some of the important concepts math majors will need in future classes. There are also foundational issues. A bright freshman can understand how the construction of the real number system proceeds, and therefore how calculus is built from the ground up, so to speak. The set of hyperreal numbers is another matter. It's really a bizarre set, being neither Archimedean nor a metric space.daveS_{April 9, 2019
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JohnnyB:
Then, for the second step, this can be divided by dx, yielding: . . .
I had trouble deriving your formula and then looked at your paper where you include this 'second step.' Can you legitimately divide a derivative by another differential and so build a true derivative form? In other words, when I see dy/dx, I think of the derivative of y with respect to x; not the derivative of y "divided" by a differential form. Likewise, d(dy/dx)/dx would be the derivative of the term in the numerator with respect to the differential in the denominator. It seems that if you actually took this derivative---instead of simply dividing, then you would get the same first term but the second term would be very different. What am I missing here? Also, the Faa di Bruno’s formula involves composite functions, but here were dealing with a straight forward function, y(x). Maybe you can elucidate.PaV_{April 9, 2019
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I don't doubt that, for those going higher up in mathematics, epsilon-delta proofs are helpful. However, even though I developed the notation using ideas from non-standard analysis, that doesn't actually get in the way of doing things "the old fashioned way". Since the form of the original second derivative was not an actual quotient anyway, I don't see how writing it the new way would be problematic. It adds potential. It doesn't remove anything. In other words, if it doesn't matter how you write it (i.e., it is mere notation, and doesn't reflect any actual quotients), then it will continue to not matter if you change the notation to a new one. If it does matter how you write it (i.e., someone might actually want to *use* it as a quotient), then my approach is clearly better. So, in the case you are referring to, the notations are neutral with respect to each other. The difference is that mine gives you the flexibility to go in other directions if you wanted to, while the old fashioned way limits you to never doing that. So, I guess I'm still not seeing any *benefit* from the old notation, though I can see some circumstances where the old notation doesn't do any active harm.johnnyb_{April 9, 2019
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johnnyb,
So, given that there is no advantage to not actually treating it as a quotient, I don’t see why we should write it as a quotient and then not use it that way.
I think there definitely are advantages for students wanting to go further in mathematics. Knowing how to write epsilon-delta proofs and more generally, working strictly in the real number system is essential for success in more advanced analysis classes, for example. For other fields, this background is less important.daveS_{April 9, 2019
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You keep insisting that dy/dx is indeed a quotient, but most of us choose not to view them that way. Therefore we need not sign on to your view that the notation for the second derivative should be obtained from dy/dx using the quotient rule.
Let me put it this way - if you do actually treat it as a quotient, then all of a sudden things that required special formulas before (such as the second derivative chain rule) can now be done by simple algebra. Additionally, you can derive an inverse function theorem for the second derivative if you treat it as a quotient. So, treating it as a quotient gives you benefits. Not treating it as a quotient gives you no additional benefits, plus, since it is written as a quotient, you run into the very real problem that someone who isn't aware that people write non-quotients in quotient form might use it that way (I've actually seen this happen in engineering). So, given that there is no advantage to not actually treating it as a quotient, I don't see why we should write it as a quotient and then not use it that way. However, I do think we are both agreed that, if you aren't using it as a quotient, Arbogast's D notation is a great improvement (see the short discussion in Section 5, though its main comparison is with Lagrangian).johnnyb_{April 9, 2019
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johnnyb, Coming back to this:
The notation for the second derivative is . However, there is not a cogent explanation for this notation. I looked through 20 (no kidding!) textbooks to find an explanation for why the notation was the way that it was.
After looking at the above linked paper, I find this confusing, since your analysis of d/dx (or d/dx( )) as an operator is completely consistent with my explanation of the meaning of d^2y/dx^2 in #4 (and hazel's earlier). Are our analyses not cogent? You keep insisting that dy/dx is indeed a quotient, but most of us choose not to view them that way. Therefore we need not sign on to your view that the notation for the second derivative should be obtained from dy/dx using the quotient rule. I do agree with your thoughts on motivation of the concept of limit before actually jumping in and calculating them. They need to understand clearly what their purpose is.daveS_{April 9, 2019
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Kairosfocus -
To get into real complexities and oddities, extend to the use of partial differentials and their rules.
Speaking of partial differentials. I actually have a paper that covers very similar ground with partial differentials. I didn't include it in this paper for a few reasons (one of which is just to limit the scope of the paper, another of which is because I still had a few kinks to work out). Now that this one is published I need to finish the other one and send it out. Unfortunately, based on how long it took to publish this one, it will probably be a year or more before anyone sees it :(johnnyb_{April 9, 2019
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DaveS and Hazel - You all might be interested in this paper of mine on the teaching method that I use for calculus, and why I find that infinitesimals allows calculus to be much more natural for students to understand. Since infinitesimals are perfectly legitimate (as non-standard analysis shows), and they are easier for students to learn (as several studies and personal experience have shown), and they allow for improved notation (as the present paper shows), it seems silly not to teach using them. Simplifying and Refactoring Introductory Calculus The title comes from the computer science term "refactoring". I had originally intended to show how different common refactoring forms from computer science played into this, but eventually just wanted to finish it so that aspect got much shorter shrift than I had originally intended.johnnyb_{April 9, 2019
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johnnyb,
As mentioned in the article, the problem that the mathematicians have had with differentials actually has very little to do with mathematics, and a lot to do with philosophy. It was claimed that infinitesimals were not a rigorous conception, but that has been shown to be false, especially with non-standard analysis (though this was just a formalization of things that were already known and done by those working with infinitesimals). Non-standard analysis just gave a more rigorous way of speaking of it.
It is a philosophical difference I suppose. I just don't see there's much to be gained in insisting that dy/dx is actually a fraction of infinitesimals, even though it is possible to do so rigorously. It's simpler and ultimately more beneficial to learn basic calculus working within the real number system exclusively (IMHO).
However, as Hazel points out, in order for integration to work, you already have to be able to manipulate dy’s and dx’s. It is better to do so with a notation that actually has robust support for the operation!
It's never necessary to think of dx or dy as infinitesimals, or to think of dy/dx as a fraction, however. Edit: Do you see how parsing dy/dx as the operator d/dx applied to the function y is advantageous? (Not to mention relatively simple and consistent with the textbooks). Second Edit: I see you do discuss d/dx as an operator in the paper you just linked to---I'll try and read it later.daveS_{April 9, 2019
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hazel,
So how do you explain the isolated dx in an integral, such as F = integral( 2x * dx) = x^2.
One way is to interpret the expression f(x)dx as a differential form (a 1-form, as described here). Another way would be think of ∫ __ dx as a unit that cannot be analyzed further. You just place the function in between, perhaps place limits on the integral sign, and go from there. Certainly your explanation of 2x * dx as representing the product of the height and width of a rectangle appeals to intuition, and people do find that helpful. I tend to think of such explanations in terms of a Riemann sum, where the the dx has been replaced by Δx, which is a real number.daveS_{April 9, 2019
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PS: We should identify that an operator transforms a pre-image function into an image function, Mathematically. Physically, we have electronic, mechanical, pneumatic, hydraulic and electronic processes that effect such operations physically. Thus, the operational amplifier's significance. A major use of such devices is in integrator chains that resolve differential equations physically.kairosfocus_{April 9, 2019
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JB, read as a fraction, the notation is indeed puzzling; it is an operator, and it acts on y = f(x), so it is reasonable to think of d^2 rather than dy*dy, more or less as your eqn 2 p. 221. The appearance as a fraction is clearly misleading in a certain sense and the odd notation has done some sweeping under the carpet . . . including bridges to physical considerations starting with Newton. I understand the appeal to the alternative D notation, and that reflects the Laplace transform, which renders it as s^2. That of course lurks behind the auxiliary equation that appears in solving differential equations. (Sometimes, I wonder if Laplace transforms should be snuck in early.) To get into real complexities and oddities, extend to the use of partial differentials and their rules. Similarly, in looking at the integral, the limit-summation approach leads to multiplying rectangular strips of width dx, so in effect it is a sum of multiples of infinitesimals, and the splitting dy/dx = f(x) --> dy = f(x)*dx --> S*dy = Sf(x)*dx and of course S*dy = y, with S standing in for elongated s, a form that has now dropped out of typography except in Mathematics. II only mention the superposed loop for integration across a closed domain such as a loop or surface to study things like fluxes. And yes, that's another thing that is glossed over too often. It seems to me, that -- as is so common -- the "simple" things used as first educational steps lurk next to deep complexities. Which then leads to glossing over. Try another: spinning a Faraday disk at right angles to a B-field generates an EMF, but spinning the magnet and imagining this spins the field relative to the disk and should yield an EMF fails. There are many sharks swimming in these waters. KFkairosfocus_{April 9, 2019
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Hazel - Your description of the current notation may be how it is intended, but it is in fact incorrect, even as you present it, which is the purpose of the paper (see the last paragraph of section 3). You said,
Now consider this: dy/dx can be thought of as the rate of change of y in respect to x. The second derivative is the derivative of the derivative, so it can be thought of as d (dy/dx)/dx: that is, the derivative of dy/dx in respect to x, again. The notation suggest that the new numerator is d(d(y) and the new numerator, as with the unit example above, is (dx)^2. Therefore, we write d^2y in the numerator to mean the derivative of the derivative, but just dx^2 (dropping the parentheses) in the denominator
While I've seen this justification before, it misses a crucial step. The problem with your suggestion is that if you actually perform "d (dy/dx)/dx", the differentiation step would have to actually perform the differentiation operation. Since dy/dx is a quotient, the proper way to differentiate a quotient is with the quotient rule. That's why the d^2y/dx^2 doesn't work. That is not the application of the quotient rule. When you apply the quotient rule, then you get my new notation. So, you (and pretty much everyone else) were on the right track, but, because you failed to recognize that dy/dx was actually a quotient, you got off at the wrong station.johnnyb_{April 8, 2019
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DaveS - As mentioned in the article, the problem that the mathematicians have had with differentials actually has very little to do with mathematics, and a lot to do with philosophy. It was claimed that infinitesimals were not a rigorous conception, but that has been shown to be false, especially with non-standard analysis (though this was just a formalization of things that were already known and done by those working with infinitesimals). Non-standard analysis just gave a more rigorous way of speaking of it. It is true that mathematicians prefer limits and physicists prefer infinitesimals. I, however, am neither. I'm a theologian with an interest in mathematics. I do agree that Arbogast's notation is preferable to what we have now, and I mention it in the paper. However, as Hazel points out, in order for integration to work, you already have to be able to manipulate dy's and dx's. It is better to do so with a notation that actually has robust support for the operation!johnnyb_{April 8, 2019
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Fun discussion, Dave. So how do you explain the isolated dx in an integral, such as F = integral( 2x * dx) = x^2. When I was first learning to teach calc so kid could understand, one of my students want to know "where did the dx go" when you integrated, and I couldn't explain. Later, when I learned to teach understanding the integral as an area, I explained that 2x was the height of an infinitely skinny rectangle, with base dx, so 2x dx stood for the area of that infinitely skinny rectangle, and then integration just added up all the rectangles to get the area. Do you buy that? :-)hazel_{April 8, 2019
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johnnyb,
The notation for the second derivative is d^2y/dx^2 . However, there is not a cogent explanation for this notation. I looked through 20 (no kidding!) textbooks to find an explanation for why the notation was the way that it was.
FWIW, here's how I understand the notation. d/dx is an operator which sends functions to their derivatives. You can compose this operator with itself. The second derivative of y wrt x is d/dx( d/dx( y ) ), which is also (d/dx o d/dx)(y), the "o" meaning composition. More briefly, this equals (d/dx)^2(y) or finally, d^2y/dx^2. It's just like how we sometimes denote a linear transformation applied twice to a vector: T(T(v)) = T^2(v).daveS_{April 8, 2019
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hazel, Regarding #2, yes those manipulations are what I was referring to. :-) Of course they have some utility, but the meaning of expressions such as an isolated "dx" is not usually accessible to a first-year calculus student. I can see how it's pragmatic to teach such things, on the other hand. #4: Yes, that's what I had in mind. And it reinforces the notion that d/dx is itself a function which sends functions to their derivatives.daveS_{April 8, 2019
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Some thoughts 1. I can give what some people might take as a cogent explanation why we use the notation we do for the second derivative, although it will be hard to type it. Also, I'm not saying that the problems johnnyb has identified aren't legitimate (I just glanced at the paper), but here is, approximately, what I used to explain to my students. (Note: this was an introductory high school course in a small school, not a rigorous college Calc I course.) First, consider the units in a simple example: if s = distance in feet, s’ = v = ds/dt is in ft/sec (which agrees with the ds/dt notation). Next s’’ = a = ft/sec/sec, or ft/sec^2. Notice that units of the denominator are squared, but those of the numerator are not. Now consider this: dy/dx can be thought of as the rate of change of y in respect to x. The second derivative is the derivative of the derivative, so it can be thought of as d (dy/dx)/dx: that is, the derivative of dy/dx in respect to x, again. The notation suggest that the new numerator is d(d(y) and the new numerator, as with the unit example above, is (dx)^2. Therefore, we write d^2y in the numerator to mean the derivative of the derivative, but just dx^2 (dropping the parentheses) in the denominator 2. To Dave: how do you feel about this? Let f = x^2. Therefore f’ = dy/dx = 2x, so dy = 2x*dx. Or how about implicit differentiation: if x^2 + y^2 = 25, can we write 2x*dx + 2y*dy = 0, so dy/dx = -x/y? Is this one of the crazy things you were referring to? :-) 3. Also, Dave, you write, “it’s (dy/dx) not a quotient. Rather, it’s just one fairly suggestive notation for the derivative of a function.” The way I taught the meaning and derivation of derivative was to take the slope of a secant line between a fixed point P and another point Q and consider the slope quotient delta y/delta x. Then let Q move towards P and take the limit of the resulting slope as the distance between P and Q becomes infinitesimally small, so that delta y become dy and delta x becomes dx, and thus dy/dx represents the “infinitely small” slope triangle at P: that is, the derivative of the function at that point. That is, a think it’s a good suggestive notation for derivative because it does represent the limit of a quotient that the students are very familiar with. I know some may think this very unrigorous but I think I had a lot of success building on these ideas to get successful calculus students. 4. And I agree about D notation, if for no other reason than efficiency: D(x^2) = 2x is nice and clean. Then you can write y' or dy/dx or whatever is convenient for 2x. Is this what you had in mind?hazel_{April 8, 2019
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