# Why isn't implicit differentiation using infinitesimals taught earlier? **URL:** \ **Category:** In My Humble Opinion\ **Created:** [March 11, 2019, 6:44am UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933 "2019-03-11T06:44:16Z")\ **Posts on this page:** 18\ **Page:** 4
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [March 12, 2019, 8:45pm UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/61 "2019-03-12T20:45:30Z")
> [@kopek](#): > > I get just enough of the discussion that I had to make some notes and do some calculations/equations myself and follow it along. If there’s anything you’d like me to elaborate on, please let me know. I’m not close to the level of 3Blue1Brown, but perhaps I can still offer something that will lead to a new insight. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [March 12, 2019, 8:53pm UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/62 "2019-03-12T20:53:50Z")
> [@Hellestal](#): > > For math profs who are tasked with teaching introductory calculus, “calculus” is simply not an independent tool, in and of itself, for use in engineering and the sciences. That is not how they see it. For people who live their lives in higher maths, calculus is interpreted as merely the students’ first taste of real analysis, dotted with real-world examples for pedagogical convenience. An interesting perspective–thanks. I have only the vaguest notions of most of real analysis: as you suggest, for me, calculus is a tool I use in in engineering and science. But math is taught by the math department and they have their own motivations. Not that I want calculus to be taught in a crank-the-handle sense, either–that’s part of the motivation behind this thread. But if calculus is going to be taught to people using it as a tool, then the foundation that’s used to build it should reflect this, and not be selected just because it’s a stepping stone to some other math. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [March 12, 2019, 9:04pm UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/63 "2019-03-12T21:04:18Z")
> [@DPRK](#): > > The very first thing you study to learn calculus is how to realize and work with the real numbers; see **Nava** ’s post for example. All I see in **Nava** ’s post is mention of limits. Limits are useful but they aren’t used (in pre-calc) to build up the real number system. Anyway, my issue with your comment was just with the word “formally”. Very little is done formally at the time (which is totally fine, of course). > [@DPRK](#): > > But the gritty details of the formal basis of the theory do not normally rear their head so that the person differentiating a function needs to worry about whether limits were used to prove a theorem versus some other technique. I agree with all this. My suggestions in this thread only apply to first-year calculus, and go away when students are comfortable with more abstract manipulations. But it’s always nice to provide a foundation that students can fall back on when other methods fail, or maybe just when they are trying to really understand some process at an intuitive level. This is a reason why I really enjoy 3Blue1Brown videos in general. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [March 12, 2019, 9:24pm UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/64 "2019-03-12T21:24:10Z")
> [@Dr.Strangelove](#): > > I’ll have to pick that up. My understanding of infinitesimal calculus comes only from what I’ve read on the internet. > > I don’t think this matters. The average student uses the reals all the time but no one but math majors ever even encounter a Dedekind cut. Same goes for set theory and Zermelo-Fraenkel axioms. At some point, we just have to accept that the structures we’re using have been proven to be consistent. I totally agree and the fact that Kiesler actually worried about the construction (though only in an appendix, I think) may have contributed. BTW, the Cauchy reals are probably a bit easier to understand than the Dedekind. There is also a construction due to Emil Artin. But none of this is or should be presented in Calc I. And I think another construction of infinitesimals would be formal Laurent series in a variable called h, ordered in such a way that h is infinitesimal. Then the reals are the power series and the ordinary part of a power series is its constant term. The main difficulty is how to extend functions to a power series. What is sin of a power series? It can be done by using the power series of sin, but is not pleasant. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [March 13, 2019, 3:03am UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/65 "2019-03-13T03:03:29Z")
> [@Hellestal](#): > > The thinking goes: You’re dealing with very small numbers. Kinda like 0.001, right? But with as many zeroes as you want to add. > > But when you multiply a very small number times itself – still very small – then the product is an extraordinarily small number. So small, it no longer matters, for example 0.001 times itself results in 0.000001. The first order terms are arbitrarily small, and yet still big enough to make a difference. But when you multiply them together, they get so small that you discard them as irrelevant. So small positive X small positive = 0? Not in an integral domain it doesn’t. OR You’re saying 1.000000000000000000000000000000000000000000000000000001 = 1? ---
**Author:** ![Hellestal](https://avatars.discourse-cdn.com/v4/letter/h/3ab097/32.png) [@Hellestal](https://boards.straightdope.com/u/Hellestal)\ **Post date:** [March 13, 2019, 3:56am UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/66 "2019-03-13T03:56:46Z")
> [@Saint\_Cad](#): > > So small positive X small positive = 0? Not in an integral domain it doesn’t. > > OR > > You’re saying 1.000000000000000000000000000000000000000000000000000001 = 1? I’m not sure what you’re after here. If you want a in-depth explanation of how infinitesimals are handled, cites have already been provided in this thread. A quick-and-dirty explanation is all you’ll get here, at least from me. An infinitesimal is non-zero, so you can divide by it without breaking math like dividing by zero would do. But simultaneously it’s small enough to act like zero in _other_ situations, so that it sometimes can be ignored. And if you multiply it by itself, the product is so super-duper small it can pretty much always be ignored. Confusing? Sure, can be. That’s the whole reason real analysis as it developed in the 19th century ignored infinitesimals and relied on limits instead – while still preserving Leibniz’s intuitive notation even as they changed the explanation behind the notation. But there’s a damn good reason infinitesimals came “first”, if not quite in rigorous form. It’s just a really natural formulation for some people, including – it should be repeated – the two guys who discovered the idea in the first place. So it’s no surprise that a lot of people even today hear this way of thinking and go “Yeah I get that” in a way they just don’t when we talk limits. But not everybody responds well to this. The foundations of calculus were pretty damn mysterious to people for quite a long time. A math prof friend of mine said the whole thing was a building made of brick with no mortar: people could see the building was standing but couldn’t see what was holding it together. It wasn’t until the 20th century that the rigorous theory of infinitesimals was finally put together. If you want the rigorous version, rather than the simple story, you’ll need to look into non-standard analysis. ---
**Author:** ![octopus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/octopus/32/3716_2.png) [@octopus](https://boards.straightdope.com/u/octopus)\ **Post date:** [March 13, 2019, 3:59am UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/67 "2019-03-13T03:59:08Z")
> [@Nava](#): > > I’m not 100% sure what techniques is the OP talking about, since I’ve never studied derivatives in English, but I think part of the issue here is that, AIUI, in the US integrals go before derivatives. That’s why you can start directly on chains: in our case, we did derivatives first. > > So for us it went: > > 1. limits. > 2. using limits to find the basic derivatives (I think this may be what the OP is proposing, only with a different name). > 3. learning combination techniques such as chaining. I’m in the US and we learned it in the same order you did and then moved on to integration. As to why implicit differentiation is taught when it is, I couldn’t answer. I’m not sure why anything is taught when it is to be honest. ---
**Author:** ![octopus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/octopus/32/3716_2.png) [@octopus](https://boards.straightdope.com/u/octopus)\ **Post date:** [March 13, 2019, 4:23am UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/68 "2019-03-13T04:23:32Z")
> [@Beckdawrek](#): > > Okay, I’ll say it. I have no clue what this is about. Jesus H.Christ this whole thread just gave a headache. So, if the SDMB were going to test infinitesimals vs limits for teaching calculus would you volunteer as a guinea pig? ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [March 13, 2019, 4:28am UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/69 "2019-03-13T04:28:33Z")
> [@BigT](#): > > I’d say it’s because the whole “common core” idea hasn’t made it up to the higher mathematics. I was thinking about this further, and think it’s a good insight. And there is a more concrete connection than one might think. I have a friend who was tutoring a kid using Common Core techniques and we’d talked about some of the strategies. One of them was expressing multiplication of sums geometrically. So for instance, one might ask what (5+2)_(3+1) is. Well, one way is to reduce it to 7_4. But another, equivalent way, is to look at the sums as sides of a rectangle and sum up the four sub-rectangles: in this example, it would be 5_3+5_1+2_3+2_1. Same answer, different method–and good for deep insight. Consider a big square, say 1000_1000. If we increase the side length by 1, how much bigger does it get? Well, one answer is 1001_1001-1000_1000=2001. But another way is to look at as four rectangles: the original 1000_1000 square, two 1_1000 rectangles, and finally a tiny 1_1 square in the corner. We add up the three _new_ rectangles and get 2001 as before. We might do the same for a 1000000\*1000000 square, which gives an answer of 2000001. It’s always double the side length plus 1. If we keep getting bigger and bigger, the difference in area grows but that little corner square doesn’t. Eventually, we can ignore it. This observation is exactly equivalent to the fact that the derivative of x[sup]2[/sup] is 2x. We have a square that’s growing on two of its edges, and there’s also a little extra corner piece but it’s so small that you can ignore it. So the derivative ends up being twice the side length. Hopefully those students exposed to Common Core techniques are able to apply this type of insight to their calculus education even without infinitesimals. ---
**Author:** ![Beckdawrek](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/beckdawrek/32/3647_2.png) [@Beckdawrek](https://boards.straightdope.com/u/Beckdawrek)\ **Post date:** [March 13, 2019, 4:59am UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/70 "2019-03-13T04:59:41Z")
> [@octopus](#): > > So, if the SDMB were going to test infinitesimals vs limits for teaching calculus would you volunteer as a guinea pig? I’m not a good guinea pig. (Cuz I’m just so unique, you know:))But what are they paying? Just so I know. ---
**Author:** ![Doctor\_Jackson](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/doctor_jackson/32/32_2.png) [@Doctor\_Jackson](https://boards.straightdope.com/u/Doctor_Jackson)\ **Post date:** [March 13, 2019, 1:07pm UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/71 "2019-03-13T13:07:26Z")
> [@Thudlow\_Boink](#): > > That’s a hijack, if not a threadshit. Sorry you think so, it was not my intent. Please see my reply in the Great Debates thread for more detail. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [March 13, 2019, 1:55pm UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/72 "2019-03-13T13:55:14Z")
> [@BigT](#): > > All you seem to be asking is why Calculus is not taught the way 3Blue1Brown does it, where he tries to make calculus more intuitive, and less about memorizing formulas and just “doing the math.” I’d say it’s because the whole “common core” idea hasn’t made it up to the higher mathematics. > [@Dr.Strangelove](#): > > Hopefully those students exposed to Common Core techniques are able to apply this type of insight to their calculus education even without infinitesimals. Actually, starting in the 1980s there was a movement to reform the way Calculus was taught. (Disclaimer: I learned math in the 1980s and have been teaching it since, and have never been in a situation where “reform calculus” was directly discussed in any detail, though I have certainly seen it referred to.) I tried, without a whole lot of success, to find a user-friendly definition or description of the Reform Calculus movement. I did come across this 1997 article: [“Reform Calculus” Has Been a Disaster, Critics Charge](https://www.math.wisc.edu/~miller/old/calc-reform.html); judging by that article, both the approach taken, and the criticisms against, Reform Calculus remind me of those of Common Core math. However, one idea that has stuck (judging from my experience) is that, in math/calculus instruction, whenever possible, concepts should be presented and looked at from multiple points of view: algebraically, graphically, numerically, and verbally. ---
**Author:** ![Nava](https://avatars.discourse-cdn.com/v4/letter/n/da6949/32.png) [@Nava](https://boards.straightdope.com/u/Nava)\ **Post date:** [March 18, 2019, 10:32am UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/73 "2019-03-18T10:32:07Z")
> [@Dr.Strangelove](#): > > I’m actually really looking forward to teaching my nephews math (though it’s some years away). I’m certain my sister will send them to me. My worry is that I will find a way of teaching that is intuitive and natural, but doesn’t correspond easily to how they learn things in school, and will make things harder initially. Whenever I’ve tutored someone (most often, in chemistry), I’ve begun by checking what and how were they taught. If we had time and they had the inclination, we went beyond the requirements - but that’s an IF. The target was to help them pass the course (often despite the teacher not knowing an electron from an elephant), not to turn them into chemists. > [@Dr.Strangelove](#): > > All I see in **Nava** ’s post is mention of limits. Limits are useful but they aren’t used (in pre-calc) to build up the real number system. Real numbers were 6th grade, limits were 10th… ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [March 18, 2019, 8:01pm UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/74 "2019-03-18T20:01:54Z")
**Doctor Jackson** , I’ll agree with your point that you can live a successful and rewarding life without calculus (as distinct from, say, algebra, which is beneficial to everyone). And that’s a good argument for not requiring calculus in schools. But, well… Calculus _isn’t_ required in schools. It’s an option, only taken by those students who have some reason to be interested in it (maybe because they’re planning on going into engineering or some other field that will use it, maybe because they want to impress college admissions boards, maybe just because they find math beautiful and worth study for its own merits). And there are certainly some people for whom calculus is useful or otherwise worthwhile. To those people, it would be a great loss if it weren’t taught at all. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [March 18, 2019, 8:15pm UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/75 "2019-03-18T20:15:09Z")
> [@Derleth](#): > > No, we definitely learn differentiation before integration in American math classes. Concur. In high school, we did differentiation first. Integration came afterwards. This would have been in 1992. I also took an accelerated calculus class (actually, I guess it would have been calculus III/multivariable calculus) in college, and the “review” portion of the class (maybe the first week-ish) also went in that order. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [March 18, 2019, 9:25pm UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/76 "2019-03-18T21:25:17Z")
> [@Nava](#): > > Whenever I’ve tutored someone (most often, in chemistry), I’ve begun by checking what and how were they taught. If we had time and they had the inclination, we went beyond the requirements - but that’s an IF. The target was to help them pass the course (often despite the teacher not knowing an electron from an elephant), not to turn them into chemists. I don’t live near my nephews, so it’s unlikely that course-specific tutoring would be very efficient. But when the time comes, I may be able to fill in certain gaps in their knowledge. I’ve forgotten just about everything I was simply _taught_. But I’ve retained almost everything I understand the principles of. When I know how to derive something, I know it forever. > [@Nava](#): > > Real numbers were 6th grade, limits were 10th… To be honest, I hardly remember what they were teaching in 6th grade math. I was reading about complex numbers and quaternions in my spare time. What exactly did you go over in 6th grade regarding the reals? I have a somewhat hard time believing it was very deep. Prove the existence of irrationals, such as showing that sqrt(2) is irrational? Cantor’s diagonalization proof? That the reals have the least-upper-bound property while the rationals do not? I’m pretty certain that US students don’t do anything like that. We had “the number line” and some other things, but nothing that could distinguish the rationals from the reals. We had spent a lot of time on the rationals, of course–fractions are a big deal in US math education. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 20, 2019, 10:51pm UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/77 "2019-06-20T22:51:03Z")
> [@Chronos](#): > > That’s not partial differentiation. > > As an example of the non-intuitiveness I mean, what’s (dx/dy) \* (dy/dz) \* (dz/dx)? Why, obviously, all of the infinitesimals cancel, and so it’s just 1. So what’s (∂x/∂y)_(∂y/∂z)_(∂z/∂x)? Why, obviously, that’s… negative 1? Where’d that negative come from? The key, of course, is that while dx/dy can be broken down to something called dx divided by something called dy, ∂x/∂y (despite looking very similar) is a symbol in itself, _not_ something called ∂x divided by something called ∂y. ∂x/∂y certainly is an infinitesimal change called ∂x divided by an infinitesimal change called ∂y. The only trouble is, we have this terrible notational convention for partial differentiation which gives the same name to different things at different times, causing all kinds of confusion. In ∂x/∂y, the ∂x represents change in x when y is allowed to change but z is held fixed. In ∂x/∂z, the ∂x represents change in x when z is allowed to change but y is held fixed. These are, alas, different things being called ∂x at different times. If x is a function of y and z, then there is a total differential dx = x(y + dy, z + dz) - x(y, z). And under ordinary conditions, this can be split by linearity of derivatives into dx = dx\_{dy = 0} + dx\_{dz = 0}, where dx\_{dy = 0} = x(y + 0, z + dz) - x(y, z), and so on. What you are calling (∂x/∂y) is then dx\_{dz = 0}/dy, but NOT the total differential dx/dy. And so on. And the expression (∂x/∂y)_(∂y/∂z)_(∂z/∂x) amounts to (dx\_{dz = 0}/dy) \* (dy\_{dx = 0}/dz) \* (dz\_{dy = 0}/dx). Of course, this wouldn’t show any nice automatic cancellation. None of the numerators match any of the denominators. But none of this is the fault of treating infinitesimals as arithmetic quantities. This is all just the fault of our terrible notation for partial derivatives having confused us into distinct infinitesimals were the same as each other. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 20, 2019, 11:15pm UTC](https://boards.straightdope.com/t/why-isnt-implicit-differentiation-using-infinitesimals-taught-earlier/830933/78 "2019-06-20T23:15:55Z")
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