Why do we teach calculus students the derivative as a limit?

Question

I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?

Something a teacher might do is ask students to calculate the derivative of a function like $3x^2$ using this definition on an exam, but it makes me wonder what the point of doing something like that is. Once one sees the definition and learns the basic rules, you can basically calculate the derivative of a lot of reasonable functions quickly. I tried to turn that around and ask myself if there are good examples of a function (that calculus students would understand) where there isn't already a well-established rule for taking the derivative. The best I could come up with is a piecewise defined function, but that's no good at all.

More practically, this question came up because when trying to get students to do this, they seemed rather impatient (and maybe angry?) at why they couldn't use the "shortcut" (that they learned from friends or whatever).

So here's an actual question:

What benefit is there in emphasizing (or even introducing) to calculus students the $h \to 0$ definition of a derivative (presuming there is a better way to do this?) and secondly, does anyone out there actually use this definition to calculate a derivative that couldn't be obtained by a known symbolic rule? I'd prefer a function whose definition could be understood by a student studying first-year calculus.

I'm not trying to say that this is bad (or good), I just couldn't come up with any good reasons one way or the other myself.

EDIT: I appreciate all of the responses, but I think my question as posed is too vague. I was worried about being too specific, so let me just tell you the context and apologize for misleading the discussion. This is about teaching first-semester calculus to students straight out of high school in the US, most of whom have already taken a calculus course in high school (and didn't do well or retake it for whatever reason). These are mostly students who have no interest in mathematics (the cause for this is a different discussion I guess) and usually are only taking calculus to fulfill some university requirement. So their view of the instructor trying to get them to learn how to calculate derivatives from the definition on an assignment or on an exam is that they are just making them learn some long, arbitrary way of something that they already have better tools for.

I apologize but I don't really accept the answer of "we teach the limit definition because we need a definition and that's how we do mathematics". I know I am being unfair in my paraphrasing, and I am NOT trying to say that we should not teach definitions. I was trying to understand how one answers the students' common question: "Why can't we just do this the easy way?" (and this was an overwhelming response on a recent mini-evaluation given to them). I like the answer of $\exp(-1/x^2)$ for the purpose of this question though.

It's hard to get students to take you seriously when they think that you're only interested in making them jump through hoops. As a more extreme example, I recall that as an undergraduate, some of my friends who took first year calculus (depending on the instructor) were given an oral exam at the end of the semester in which they would have to give a proof of one of 10 preselected theorems from the class. This seemed completely pointless to me and would only further isolate students from being interested in math, so why are things like this done?

Anyway, sorry for wasting a lot of your time with my poorly-phrased question. I know MathOverflow is not a place for discussions, and I don't want this to degenerate into one, so sorry again and I'll accept an answer (though there were many good ones addressing different points).

Answer 1

This is a good question, given the way calculus is currently taught, which for me says more about the sad state of math education, rather than the material itself. All calculus textbooks and teachers claim that they are trying to teach what calculus is and how to use it. However, in the end most exams test mostly for the students' ability to turn a word problem into a formula and find the symbolic derivative for that formula. So it is not surprising that virtually all students and not a few teachers believe that calculus means symbolic differentiation and integration.

My view is almost exactly the opposite. I would like to see symbolic manipulation banished from, say, the first semester of calculus. Instead, I would like to see the first semester focused purely on what the derivative and definite integral (not the indefinite integral) are and what they are useful for. If you're not sure how this is possible without all the rules of differentiation and antidifferentiation, I suggest you take a look at the infamous "Harvard Calculus" textbook by Hughes-Hallett et al. This for me and despite all the furor it created is by far the best modern calculus textbook out there, because it actually tries to teach students calculus as a useful tool rather than a set of mysterious rules that miraculously solve a canned set of problems.

I also dislike introducing the definition of a derivative using standard mathematical terminology such as "limit" and notation such as $h\rightarrow 0$. Another achievement of the Harvard Calculus book was to write a math textbook in plain English. Of course, this led to severe criticism that it was too "warm and fuzzy", but I totally disagree.

Perhaps the most important insight that the Harvard Calculus team had was that the key reason students don't understand calculus is because they don't really know what a function is. Most students believe a function is a formula and nothing more. I now tell my students to forget everything they were ever told about functions and tell them just to remember that a function is a box, where if you feed it an input (in calculus it will be a single number), it will spit out an output (in calculus it will be a single number).

Finally, (I could write on this topic for a long time. If for some reason you want to read me, just google my name with "calculus") I dislike the word "derivative", which provides no hint of what a derivative is. My suggested replacement name is "sensitivity". The derivative measures the sensitivity of a function. In particular, it measures how sensitive the output is to small changes in the input. It is given by the ratio, where the denominator is the change in the input and the numerator is the induced change in the output. With this definition, it is not hard to show students why knowing the derivative can be very useful in many different contexts.

Defining the definite integral is even easier. With these definitions, explaining what the Fundamental Theorem of Calculus is and why you need it is also easy.

Only after I have made sure that students really understand what functions, derivatives, and definite integrals are would I broach the subject of symbolic computation. What everybody should try to remember is that symbolic computation is only one and not necessarily the most important tool in the discipline of calculus, which itself is also merely a useful mathematical tool.

ADDED: What I think most mathematicians overlook is how large a conceptual leap it is to start studying functions (which is really a process) as mathematical objects, rather than just numbers. Until you give this its due respect and take the time to guide your students carefully through this conceptual leap, your students will never really appreciate how powerful calculus really is.

ADDED: I see that the function $\theta\mapsto \sin\theta$ is being mentioned. I would like to point out a simple question that very few calculus students and even teachers can answer correctly: Is the derivative of the sine function, where the angle is measured in degrees, the same as the derivative of the sine function, where the angle is measured in radians. In my department we audition all candidates for teaching calculus and often ask this question. So many people, including some with Ph.D.'s from good schools, couldn't answer this properly that I even tried it on a few really famous mathematicians. Again, the difficulty we all have with this question is for me a sign of how badly we ourselves learn calculus. Note, however, that if you use the definitions of function and derivative I give above, the answer is rather easy.

Answer 2

I'm teaching Calc 1 this semester, and I've stumbled onto something that I like very much.

First of all, I start (always) by having my students draw bunches of tangent lines to graphs, compute slopes and draw the "slope graphs" (they also do "area graphs", but that's not relevant to this answer). They build up a bit of intuition about slope and slope graphs.

Then (after a few days of this) I ask them to give me unambiguous instructions about how to draw a tangent line. They find, of course, that they are stumped.

In the past, I went from this to saying "we can't get a tangent line, but maybe we can get an approximately tangent line" and develop the limit formula.

This semester, I said, "we have an intuitive notion of tangency; suppose someone offered a definition of tangency -- what properties would it satisfy?" We had a discussion with the following result: tangency at point $x = a$ should satisfy:

1. tangency (of one function with another) should be an equivalence relation
2. if two linear functions are tangent at $x= a$, they are equal.
3. a quadratic has a horizontal tangent line at its vertex.
4. if $f$ and $g$ are tangent at $x = a$, then $f(a) = g(a)$.
5. if $f_1$ is tangent to $f_2$ at $x = a$ and $g_1$ is tangent to $g_2$ at $x = a$ then $f_1 + g_1$ is tangent to $f_2 + g_2$ at $x = a$ and similarly for the products.
6. the evident rule for composition.

Using these rules, we showed that if $f$ has a tangent line at $x = a$, it has only one. So we can define $f'(a)$ to be the slope of the tangent line at $x = a$, if it exists!

The axioms are enough to prove the product rule, the sum rule and the chain rule. So we get derivatives of all polynomials, etc., assuming only that tangency can be defined.

Then (limits having presented themselves in the computation of area) I defined $f$ to be tangent to $g$ if $\lim_{x\to a} {f(x) - g(x) \over x-a} = 0$. We derive the limit formula for the derivative, and check the axioms.

EDIT: Here's some more detail, in case you're wondering about implementing this yourself. I had the initial discussion about tangency in class, writing on the board. A day or so later, I handed out group projects in which the axioms were clearly stated and numbered, and the basic properties (as outlined above) given as problems.

The students' initial impulse is to argue from common sense, but I insisted on argument directly from the axioms. There was one day that was kind of uncomfortable, because that is very unfamiliar thinking. I had them work in class several days, and eventually they really took to it.

Answer 3

I'm going to answer this part:

does anyone out there actually use this definition to calculate a derivative that couldn't be obtained by a known symbolic rule?

Yes. $sin(x)$.

My point is that of course we can just learn the derivative of this function, but then we could just learn the derivative of any function. So looking for a "complicated function" that needs the limit definition is pointless: we could just extend our list of examples to include this function. It's a bit like the complaint that there's no closed form for a generic elliptic integral: all we really mean is that we haven't given it a name yet.

In fact, one could do $x^2$ like this, or even $x$, but I think that $sin(x)$ has a good pedagogical value. If you can get them first to ponder the question, "What is $sin(x)$?" then it might work. I'm teaching a course at the moment where I'm trying to get my students out of the "black box" mentality and start thinking about how one builds those black boxes in the first place. One of my starting points was "What is $sin(x)$?". Or more precisely, "What is $sin(1)$?". If you take that question, it can lead you to all sorts of interesting places: polynomial approximation of continuous functions, for example, and thence to Weierstrass' approximation theorem.

Many students will just want the rules. But if the students refuse to learn, that's their problem. My job is to provide them with an environment in which they can learn. Of course, I should ensure that what they are trying to learn is within their grasp, but they have to choose to grasp it. So I'm not going to give them a full exposition on the deep issues involving the ZF axioms if all I want is for them to have a vague idea of a "set" and a "function", but I am going to ensure that what I say is true (or at the least is clearly flagged as a convenient lie).

Here's a quote from Picasso (of all people) on teaching:

So how do you go about teaching them something new? By mixing what they know with what they don't know. Then, when they see vaguely in their fog something they recognise, they think, "Ah, I know that." And it's just one more step to, "Ah, I know the whole thing.". And their mind thrusts forward into the unknown and they begin to recognise what they didn't know before and they increase their powers of understanding.

We all remember professors who forgot to mix the new in with the old and presented the new as completely new. We must also avoid the other extreme: that of not mixing in any new things and simply presenting the old with a new gloss of paint.

Answer 4

While I think that ideally, even in a freshman course of calculus, students should receive some historical notions about the development of the ideas of infinitesimal calculus, I think that, even in a freshman course of calculus, the true definition of derivative of a function should be given, that is, via the first order approximation. A function $f:(a,b)\to\mathbb{R}$ is differentiable at $x$ if there exists $m$ such that $$f(x+h)=f(x)+mh+o(h)\quad \mathrm{as}\\ \\ h\to0. $$

The fact that the coefficient $m$ (the derivative) can be characterized, and sometimes efficiently computed, as a limit of a quotient, has certainly to be observed, and should be applied immediately to treat some elementary functions like $x^2$, $1/x$ or $e^x$, as usual. But I would never give it as a definition.

I think there is a philosophical issue here. It may seem simpler to define something as the result of a procedure for getting it, compared with defining it via a characteristic property. But the latter way is superior, and on a long distance, simpler. And in the case of students who will stop there their mathematical education, then, I prefer they at least see the true idea behind, rather that being able to compute the derivative of $\cos(e^x)$ : when will that be of use for them?

The definition via first order expansion is very natural, and more understandable to the freshman students. It has a more direct geometrical meaning. It reflects the physical idea of linearity of small increments (like in Hooke's law of elasticity, etc). It is much closer to the practical use of derivatives in approximations. It makes easier all the elementary theorems of calculus (consider how needlessly complicated becomes the proof of the theorem for the derivative of a composition by introducing a useless quotient). Finally, it is closer to the generalization to Fréchet differential, which is a good thing for those students that will continue their study in maths.

A funny remark, from my experience. Ask students that received the definition of derivative as limit of incremental quotient, to compute $\lim_{x\to 0 }\sin(x)/x$. Will anybody say, it's the derivative of $\sin(x)$ at $0$, that is $\cos(0)=1$? No, they will try and use the "rule of de L'Hopital"!

Answer 5

I agree with the above comments.

The point of my comment-question "What competing definition do you have in mind?" was to emphasize something that seems to be under-emphasized in the question itself: the reason we speak of derivatives as limits is because that's the definition of the derivative, and we want to give a definition of the concept that is going to be discussed for much of the semester.

[It is possible to give other definitions of a derivative, but they are all variations on the same theme and, in particular, all use either the concept of limit or the (equivalent!) concept of continuity. For instance, Caratheodory has a nice definition of the derivative in terms of functions vanishing to first order, but this is not going to be any more palatable to the freshman calculus student.]

[Added: I admit that I forgot about nonstandard analysis when I wrote the above paragraph. That indeed has a somewhat different feel from the usual limits and continuity. One the one hand, although I have never taught calculus this way, I rather doubt that doing so would suddenly make the difficult concepts of continuity and differentiability go over easily. On the other hand, I certainly couldn't decide to teach a nonstandard approach to calculus because it would be...nonstandard. The curriculum among different sections, different classes and different departments has to have a certain minimal level of coherence, and at the moment the majority of the grad students and faculty in every math department I have ever seen are not familiar enough with nonstandard analysis to field questions from students who have learned calculus by this approach.]

If we don't give a definition of the most important concept in the course, then we lose all pretense of developing things in a logical sequence. In particular, it's hard to see how to discuss the derivations of any of the basic rules the students will actually be using to compute derivatives, and thus we would be forced to reduce calculus to a (long!) list of algorithms based on certain unexplained rules.

Nevertheless I take your question seriously, since I have taught a fair amount of freshman calculus in recent years. It is absolutely correct that a lot of students get impatient, angry and/or confused at the limit definition of the derivative (or really, at anything having to do with limits and/or continuity). I do derivations of things like the product rule and the power rule rather quickly in class, because I know that something like half the class isn't following and doesn't care to follow. And yet I do them anyway (not all of them, but more than half) because, to me, not to do them makes the course something I could not bring myself to teach (and, by the way, would put it well below the level of the AP calculus class I had in high school: I feel honorbound to give to my calculus students not too much less than was given to me). Thus there is a real disconnect between the calculus class that I want to teach and the calculus class that something like half of the students want to take. It's discouraging.

I would be happy to hear that I am making a false dichotomy between giving the limit definition of the derivative and just giving algorithms to solve problems. I definitely experiment with different kinds of explanation beyond (and instead of!) just a formal proof. Here are some things I have tried:

1) Take the definition of continuity as primary, and define the limit of a function at a point as the value at which one can (re)define the function to make it continuous. I think this should be helpful, since I think most people have an intuitive idea of a "continuous, unbroken curve" and much less of the limit of a function at a point.

2) Emphasize physical reasoning. The last time I taught freshman calculus, I spent the entire first day talking about velocities: first average velocity, then instantaneous velocity. If a differentiation rule has a plausible physical interpretation -- e.g. the chain rule says that rates of change should multiply -- then I often give it.

3) Emphasize "chemical reasoning", i.e., dimensional analysis:. I often give the independent variable and the dependent variable units and emphasize that the units of the derivative are different from the units of the original function. In this way one can see that the conjectured product rule $(fg)' = f'g'$ is dimensionally wrong and thus nonsense. (And again, the chain rule is "obvious" from a unit conversion perspective.) Similarly dimensional analysis should stop you from saying that the volume of a cylinder is $\pi rh$.

Unfortunately none of these things have worked with the portion of the class that doesn't want to hear anything but how to solve the problems.

Added: To more directly address your specific question: yes, there are problems one can ask of freshman calculus students which require them to use the limit definition of the derivative rather than (just) the differentiation rules, but I do not recommend asking many of these questions, since the students find them very difficult. A personal example: when I was teaching Math 1A (first semester calculus) as a graduate student at Harvard, we had communal exams but the course head (who was a tenured professor of mathematics, hence a very brilliant person) had the final say. On the first exam, we decided that one of the questions was too hard, so at the last minute the course head replaced it with the following one (which he did not show to us):

Consider the function $f(x)$ defined as $x^a \sin(\frac{1}{x^2})$ for $x \neq 0$ and $f(0) = 0$. What is the smallest integer value of $a$ such that $f$ is (i) continuous, (ii) differentiable, (iii) twice differentiable?

I had the good fortune to grade this problem. Out of $200$ or so exams, the median score was $0.5$ out of $12$. About three students wrote down the right numerical answer for part (iii), but this was not supported by any work or reasoning whatsoever.

Added: by the way, it's not as though the above question is "bad" in the sense that it's not testing mathematical competence and depth of understanding of calculus. I think it absolutely is, just at a level way above that which one should be testing in a freshman class for non math majors. For the next few years, when the story came up in a social setting involving mathematical hotshots, after telling it I would press them for an answer to part c) on the spot. Most people I asked did not get it. (Note that I would not of course give them pen and paper and a quiet spot to think about the problem for some period of time. I generally required an answer after a minute or so. Let's hold PhD mathematicians to higher standards than freshman non-majors after all!) For instance, I watched a cloud pass over one Fields Medalist's face as he got very confused. After a while though I stopped using this as a pop quiz in addition to a story: I can't explicitly remember why, but I'd like to think it dawned me how obnoxious it was to put people on the spot like that...

Answer 6

I wanted to add one further point to the many good answers already given here: "black box" symbolic computation, in the absence of understanding the formal definitions, can work when everything goes right, but is very unstable with respect to student errors (which are sadly all too common). Knowledge of definitions provides a crucial extra layer of defence against such errors. (Of course, it is not the only such layer; for instance, good mathematical or physical intuition and conceptual understanding are also very important layers of defence, as is knowledge of key examples. But it is a key layer in situations which are too foreign, complicated, or subtle for intuition or experience to be a good guide.)

For instance, without knowing the formal definition of the derivative, a student could very easily start with a true formula such as

$$ (x^2)' = 2x$$

and do something like "substitute x=3" to obtain the false formula

$$ (9)' = 6.$$

(An example that I have actually seen: someone attempted to prove Fermat's last theorem by starting with the equation

$$ a^n + b^n = c^n$$

and then differentiating with respect to $n$. Ironically, a variant of this type of trick actually works when solving FLT over polynomial rings, but that's another story...)

Now, without bringing in the definition of a derivative (and of a function), how could you explain to the student what went wrong here in a way that the student will actually remember? Saying that one can use the law of substitution or the trick of differentiating both sides in some situations, but not in others, is likely to be recalled inaccurately, if at all (and may have the side effect that the student may view such basic moves as substitution as somehow being "suspect", thus avoiding it in the future).

Answer 7

The derivative of $x|x|$ is best computed via the "limit" definition. A more general example is $xf(x)$ where $f$ is any continuous function, and we are computing the derivative at $x=0$.

Answer 8

I am surprised that no answer has explicitly mentioned the fundamental theorem calculus yet: that is a classic, and important, instance of calculating the derivative using the limit definition. So, for example, the integral sine function

$$\int_0^x \frac{\sin t}{t} dt $$

has important applications in signal processing and the cumulative distribution function of the normal distribution $N(a,\sigma^2)$

$$\frac{1}{\sqrt{2\pi\sigma^2}}\int_{-\infty}^x e^{-\frac{(t-a)^2}{2\sigma^2}}dt$$

is the bread and butter of probability and statistics. Both functions are not elementary and their derivatives, while significant, would be impossible to calculate by other means.

I also disagree with the comment that piecewise defined functions "are not good at all" for illustrating the definition of the derivative based on limits. In fact, piecewise polynomial functions, in the form of splines, are used in mechanical engineering (e.g. to design the shape of the car body), and provide a neat opportunity to relate conceptual and computational aspects of derivatives.

Answer 9

The way that Calculus is traditionally taught gives a false impression that every function worth looking at can be differentiated using the rules of differentiation. This comes from a misconception that any function worth looking at can be described by an algebraic formula, or using trigonometric or logarithmic functions.

That's just not the case: the most common everyday functions don't have any formulas. Some examples:

1. Price of a company stock over several decades.
2. Volume of water in a water tower over the course of a week
3. Median price of a house in your area (adjusted for inflation), over the course of 100 years.
4. US National Debt over the last two hundred years.
5. US Deficit

For such functions, rate of change has a very real meaning. I find that students who had Calculus in high-school are stumped if I give them an example like that and ask them to graph the rate at which, say, the US national debt has changed throughout US history, and how that relates to the deficit.

Understanding the derivative as both rate of change and the slope of the tangent line helps, and the only good way to tie those concepts is with using limits.

Answer 10

The definition of derivatives is useful in exercises about functional equations. Ever solve $f(x+y)=f(x)f(y)$ ? A more elaborate one is $[f(x):f(y):f(z):f(t)]=[x:y:z:t]$, functions preserving the cross-ratio (= anharmonic ratio).

However, we should not neglect the interest of the black-box side of mathematics. We should remember that it is this aspect which has made mathematics so much unavoidable in Science. Somehow, it participates to the ``unreasonable effectiveness of Mathematics in the Natural Sciences'' (E. Wigner's famous statement). After all, the definition of derivatives has the same status as the constructions of ${\mathbb Z},{\mathbb Q},{\mathbb R},{\mathbb C}$. One can spend a year without thinking about them, while using these fundamental objects every hour, by applying rules. Do you remember the construction of the polynomial algebra $k[X]$ ? How would you define $\pi$ ? In a more advanced situation, chemists have efficient rules to deal with characters of representations of finite groups, and they do not need to read a justification, or to remember it, even though the first Chapter of J.-P. Serre's book was intended to be read by his chemist wife. Mathematics is the tool box of Science. It is even a tool box for itself, in the sense that new topics use the older ones. To go further, we must accept older truths. Of course, it is way better to accept them for good reasons, that is, because we have completely understood the definitions. But if the half of a classroom, who does not intend to do mathematical research, neglects the definition and prefer focussing on the rules, there is no problem at all, provided they apply the rules correctly. There are many ways to learn rules, one of them being solving a lot of exercises.

Answer 11

An example I like is $\exp(-\frac{1}{x^2})$ and the "bump functions" one can construct with it.

First of all, this example is important in differential geometry (e.g. Whitney's embedding theorem) and complex analysis (as an example of a real $C^\infty$ function which isn't holomorphic).

In second place, even in first year calculus it's an important illustration of the concept of derivative and of Taylor's theorem. It's important in my opinion to understand why all derivatives at zero are zero (i.e. because it goes to zero faster then any polinomial) but even so the function is changing values.

Answer 12

Since this is community wiki, I'll feel free to share a possibly relevant anecdote; feel free to delete if you don't think this is an answer.

I once had a freshman calculus student ask me if they'd be required to learn the "Greek method" for calculating derivatives. When I looked puzzled, he explained to me that the "Greek method" involved taking a limit as $h$ goes to zero, and that this was the method the ancient Greeks had used back before anyone realized that all you have to do is bring down the exponent and then lower it by one.

Answer 13

This is a question that I also struggle with sometimes. On the one hand, I understand the value of sweeping things under the carpet when students are not ready for them yet. When I learned Calculus in High School, we talked about -- but never properly defined -- limits (I'm can't recall if we did the limit derivatives). Yet, we managed to go pretty far into the material, e.g. establishing recurrence relations for integrals of the type $\int_a^b e^{-x}\sin(n x)$. This lack of definition was a very frustrating point for me, and when I finally learned about $(\epsilon,\delta)$ two years later, a wave of relief washed over me. Yet, I'm pretty sure that my cohorts did not feel the same way, hence my sympathy for teachers who want to keep things simple by hiding the definition.

At the same time, I don't want my Calc course to be a series of magic tricks, so I always insist on the logical construction of the course: we want to investigate slopes of tangents. We want to work exactly, not approximately. This is why we'll get into limits in the first place (not very historical, but a logical development). So what do I do?

• I briefly cover $(\epsilon,\delta)$ without really applying it. Just to show the difference between a "wordy" definition and a mathematical one.
• I insist on the fact that the limit laws are derived fro this rigorous definition. (You can sketch the proof for the sum of limits for instance).
• This sets up for students how the mathematical edifice is built: abstract definitions to formalize intuition, big gun theorems proved rigorously from these definition (limit laws, derivative laws...). Add a few examples to the mix and then you're set up for practical, mechanical computations (the stuff that computers do).
• I am upfront about the fact that I don't expect my students to use the $(\epsilon,\delta)$ definition, though I like them to memorize it. The payoff will be later.
• I also stress that, historically, calculus was done without this definition for a long time: so it can be done, they will be able to do it, but it also has its limitations when dealing with more abstract material.

In a course that is set up in this way, it is quite natural to cover the limit definition of derivatives. There are a lot of good reasons why one should do that anyway, some of which have already been addressed. Functions which are defined piecewise do require this, and that includes important examples like $\exp(-1/x^2)$ and fun ones like $x^2\sin(1/x)$. The rigorous derivation of the derivative of $\sin x$ is another good example.

There are also wrong ways of doing this. In the comments, Holger pointed to the case study in the Notices article Teaching mathematics graduate students how to teach. Here, the problem asked to use the definition of derivative to compute the slope of a certain cubic at a point. By the time the exam rolls around, you have easier ways of doing this, so of course the students would feel that this is an arbitrary and confusing question.

[Actually, I took so long to write this that I've been ninja'd by Pietro on this example.] One example that I have yet to see though is Taylor series: defining the derivative in this way makes it obvious that $$ f(a+h) \approx f(a)+f'(a)h+o(h)$$ and sets you up for the higher order ones. Yes, you can see that from the graph too, but at that level most of my students have a terrible time reasoning from an abstract graph.

Given how fundamental these ideas are, especially in Physics, I can never stress enough these kinds of relationship in my course.

Answer 14

I hope my answer is read as a response to the question asked, rather than as either a defense of or disagreement with the choices the pedagogists (is that a word?) make.

I think one of the main reasons to teach derivatives in terms of the $h\to 0$ limit is that it captures the dual notions of "instantaneous velocity" and "slope", which are respectively physical and geometric.

(Ok, now I will mention some personal opinions about teaching calculus. I love physics, and sometimes pretend to be a physicist, so for me the geometric/physical meanings of calculus are very important. So I would love if they were emphasized more. Unfortunately, we do not do enough in introductory calculus classes in that direction, and it is very hard to present functions and ask students to find the slopes of their graphs without essentially teaching them these black-box techniques. So I don't know whether it's worth it: maybe we should just do the algebraic part of calculus --- it's the only thing we tend to test anyway. I also don't really think that MO is the best place to get into that discussion, though, and I don't think that OP intended as such.)

Answer 15

So far no one's mentioned (or did I miss it?) that if you make students compute $$ \lim_{w\to 5} \frac{w^6 - 5^6}{w - 5}, $$ then some of them will use L'Hopital's rule to do that, if you don't tell them not to.

Here's an example of something I have students do with the limit definition of the derivative: http://wnk.hamline.edu/~mjhardy/1170/notes/quiz.10.19.pdf

They find all sorts of creative ways of getting things wrong when doing this. Here's another: http://wnk.hamline.edu/~mjhardy/1170/homework/13th.pdf

I think after they've done several like this, they actually do learn what this is for, and that it's not being used as a way to avoid quick and efficient ways of computing derivatives.

But I have them thinking about instantaneous rates of change without using limits on the first day of the course: http://wnk.hamline.edu/~mjhardy/1170/handouts/September.8.pdf

Answer 16

Well, the definition of derivative is probably one of the best application of the notion of limit, from a didactical point of view. If you define the derivative as a limit process then students who understand it will not miss the geometric flavour: the slope of the tangent line is the limit as $h\rightarrow 0$ of the slope of the line through $(x, f(x))$ and $(x+h, f(x+h))$. I think this is beautiful and relatively simple, once you get the students to think about it for a minute. Plus, it answers the question "When do we agree that the graph of $f$ admit the existence of a tangent line at $(x, f(x))$"?

Of course one has to keep in mind that for most students the useful thing to learn is how to compute practically a derivative without using the definition but rather applying a collection of rules. Nevertheless I think it is important to give them an idea of where all these rules come from. Think about those students who want to get a a math major? No?

In Italy in the so called "scientific high school", the schools that provide you with the widest and most basic education (you learn a bit of everything) with a focus in math, physics, chemistry perhaps, ecc.. we are taught the limit using the $\epsilon-\delta$ definition, and the derivative from its definition. This is to say that I think it is possible to have students learn this theoretical aspects of calculus, if high school kids do.

Answer 17

A problem I like to give students to solve shortly after introducing the derivative is to evaluate $f'(2)$ for $f(x) = x^x$. Of course, this function can be rewritten as $f(x) = e^{x\ln x}$ but in my experience students don't think of this. In fact, students who have seen Calculus before almost universally reach the solution $f'(2) = 4$ which they get from the mistaken idea that $f'(x) = x\cdot x^{x-1} = x^x$. The only students that usually get this problem correct are those that haven't yet learned any of the computational methods and only know the definition.

I teach the limit definition and emphasize the physical and geometric interpretations, and then move from that to the concept of the tangent line and linear approximation. I think these concepts encapsulate most of what is significant (intuitively) about the definition. I dislike exam questions that require students to compute derivatives using the limit definition when they know a "better" way to do it. It isn't too hard to write a problem where no formula for the function is given and ask students questions about the sign or approximate magnitude of the derivative or whether or not the function should even have a derivative. For students who to do not intend to pursue mathematics, this seems appropriate to me. Even those who become mathematicians will almost surely see these ideas again in complete detail in an elementary analysis course.

Answer 18

The standard definitions of limit, continuity and derivative are things of beauty mathematically - flexible and well-honed like fine woodworking tools. But to get calculus students to care, and appreciate their meaning and significance, takes some motivation.

A pretty good way to motivate $\epsilon$-$\delta$ is that it has to do with determining what control on input error ($\delta$) is needed to guarantee meeting a given tolerance for output error ($\epsilon$). How accurately do you have to aim a spacecraft to ensure it enters Martian orbit without burning up the way Beagle 2 did, costing hundreds of millions? Students can appreciate this is a serious question, and that it is fair to insist they be able to handle simple examples like $f(x)=-100x+50$, $\epsilon=10^{-2}$. (In large lectures for freshmen, I wouldn't do much more than Lipschitz examples or something carefully designed so $\delta$ is easy to find without cases. Many calculus students are adults but, ahem, need practice with inequalities.)

One can tell engineering students who just want the formulas that they'll be surprised to find that in a couple of years they'll be estimating ``sensitivity coefficients'' numerically from black-box software or experiment. Gee, sensitivity coefficients are just derivatives, and they'll be estimated from the definition, not symbol-pushing.

Speaking of which, it's nice to express the error in the definition explicitly: $$ \frac{f(x)-f(a)}{x-a} = f'(a)+E_a(x), $$ and do the algebra that occurs to few to write $$ f(x)=f(a)+ f'(a)(x-a)+ E_a(x)(x-a) $$ This makes the nature of linear approximation a bit more apparent.

Answer 19

One way to avoid limits without losing too much is to teach the calculus of finite differences. Conceptually, the move from numbers to lists-of-numbers as first-class mathematical objects is easier than the move from numbers to real-valued-functions-of-a-real-variable, and the easier move also forms a good stepping stone to the harder one. One can develop the calculus of finite differences mutatis mutandis and thereby make the transition to infinitesimal calculus essentially painless. (So, for example, one should work not with polynomials per se, but with linear combinations involving rising or falling powers).
All the black box rules have their analogues, and all are reasonably easy to see and/or prove. Passing the limit, when it happens, comes as a welcome simplification.

Aside from the conceptual challenge of functions themselves, students find limits difficult because of their quantifier complexity. I have never understood why standard algebra pedagogy suppresses quantifiers, thus, for example, leaving many students unable to distinguish between unknowns (literals bound by existential quantifiers), variables (literals bound by universal quantifiers) and constants (literals that belong to the language itself). Students who miscalculate the derivative of $\pi^2$, mentioned elsewhere, don't get this distinction. People who become mathematicians usually "got it" without anyone spelling all this out, and then they learned about quantifiers studying logic in college, so they regard quantifiers as sophisticated and advanced. But most students don't "get it," and I think this accounts for the huge attitude downturn when they get to algebra.

Answer 20

It's funny that actually many students believe that the symbiosis is always the other way around, i.e. derivatives are used to compute limits (l'Hopital etc.). My favorite example of an elegant calculation of a derivative using the limit definition comes from basic physics. Ask the students why the acceleration of an object performing uniform circular motion is always perpendicular to the velocity. One could come up with a non-elegant solution by writing the equations of motion and using a derivatives table, or one could observe the nice geometric proof of considering an infinitesimal isosceles triangle formed by the two velocity vectors that are a few seconds apart and notice that $\Delta \vec{v}$ is the base of this triangle and points toward the center of the circle.

Answer 21

$\frac{\sin x}{x}$ at $x = 0$ should be a good example.

P.S.: Talking about esoteric definitions, if you can introduce stationary point without derivatives, you can then introduce derivatives using sheaves, like you would introduce vectors on a smooth manifold. It would broaden the consciousness of your freshmen, he-he ^_^

Answer 22

I believe it gives a good conceptual or practical reasoning to why we would want to study calculus in the first place. As it was first introduced to me, we can always talk about the average speed a car has traveled over a certain distance. Even very small distances, but apart from a speedometer, how can we say 'I'm travelling XX km/h right now'. There enters the limit definition, where we want the instantaneous rate of change!

If we only presented the formal rules for differentiation, we run in to the same problem as high school students who dislike math present "But my calculator can just do it! Why do I need to learn this?!". If the fundamentals are not taught, one day they will be forgotten.

There are certainly other rigorous approaches to the derivative out there. The delta-epsilon method, which most students in their first year simply struggle to grasp as easily as the $h \rightarrow 0$ definition. This approach is typically reserved for the math majors who go on to take a course in analysis, not the general first calculus course for all science majors.

While I do not use this definition in practice, I am primarily not calculating derivatives, so take that for what it's worth I suppose.

Answer 23

The answer I give my students is that mathematicians want to know what a word (in this case 'derivative') means in all cases, and the definition of the derivative is a communal agreement about what to say in strange cases such as the absolute value function. (Well, since I banish symbolic stuff from the first two weeks, I say 'function whose graph has a sharp corner like this one (draws on board)'.)

If students press further, I point out that in a literature class they are expected to learn the communal agreement on the difference between a 'simile' and a 'metaphor'. It helps that I am at a liberal arts institution and not a technical one.

Let me also use this opportunity to share a pedagogical trick:

I find it helpful (third time I've tried it) to break up the definition of $f^\prime(2)$ into two parts:

1) Define a new function $E_2$ by the formula $$E_2(x)=\frac{f(x)-f(2)}{x-2}.$$ 2) Take the limit of $E_2$ at 2.

To pull this off, you do need to take the function $E_2$ somewhat seriously; graph it, write formulas for it, et c.

Rationale:

1) It always helps to break up complicated definitions into smaller pieces.

2) It emphasizes that you take limits of functions (in the sense of machines that accept a single number as input and gives a single number of output) rather than of symbolic expressions.

3) Students get to really understand why a discontinuous function or something like the absolute value function is not differentiable (at the relevant point).

Answer 24

It is worth noting that there is a lot of historical precedent for teaching it as a limit, which occurs already in Euclid. I.e. Euclid characterizes the tangent to a circle as the unique line such that between it and any other line through the same point, one can interpose a secant (Prop. 16, Book III). (Strictly, he says equivalently that one cannot interpose another line between the tangent and the circle itself, i.e. every other line through the point is a secant.) Thus the tangent is the limit of those secants. Thus I believe one can easily say that the limiting point of view is the original one of Euclid. From this point of view, the idea of limit is the one used so fruitfully by the Greeks, and the contribution of the mathematicians of later times is to make that notion more precise.

On the other hand, if you want to avoid the conceptual difficulty students have with limits, you can follow Descartes instead, at least for derivatives of polynomials, and characterize the tangent line as the unique line such that subtracting its equation from the original function gives a polynomial with a double root at the given point. This leads to motivating the Zariski cotangent space, as M/M^2.

Both points of view also have a nice dynamic interpretation as realizing the tangent as the unique line intersecting the curve doubly at the point, understood as the limit of the two secant intersections,and measured by the presence of a double root.

But if you want a defense of limits, I suggest Euclid Prop. 16, Book III as ample precedent.

If you want a defense of making students practice using the limit definition, I propose that as noted above, this is the only way to get them to appreciate the fundamental theorem of calculus. That theorem cannot be appreciated by memorizing rules for derivatives, One must understand the definition and apply it to an abstractly defined area function. I suggest that one reason many students do not understand why the fundamental theorem of calculus is true, is that (again as noted above) they have not grasped either what an abstractly defined function is, nor what a derivative truly means.

So if you want them to understand the relation between the derivative and the integral, then I agree with others that they need to know what a function is and derivative is. The reasoning here is that once someone understands something, he can use it in more settings than could possibly be covered by any set of rules.

Another practical benefit of testing the use of the h-->0 definition to obtain derivatives of simple functions, is that it forces practice in algebra, trig identities, and exponentials, skills which most of my students are almost completely lacking.

However, I recommend you teach it any way that makes sense to you. after all you understand it, so whatever you say based on that understanding will be useful. Make up your mind what seems important to you, and go for it!

Answer 25

If it is just a question of definition but not a question of computation, I have heard when I was a student the following definition:

Let $f$ be a real continuous function, class $C^0$. Let $$ \Delta f(x,x') = \frac{f(x') - f(x)}{x'-x} \quad\mbox{defined on}\quad {\bf R}^2 -\{x=x'\} $$ If $\Delta f$ admits a continuous extension on the diagonal $\{x=x'\}$ then it is unique, and $f$ is said to be of class $C^1$. The function $f'x) = \Delta f(x,x)$ is then called the derivative of $f$

Of course this is the standard definition, nothing new under the sun, but the $\epsilon-\delta$ calculus if hidden, and of course, not for long :-)

Answer 26

Another alternative way of teaching calculus is via infinitesimals (for example the book Elementary Calculus, An Infintesimal Approach by Keisler). The way of thinking about calculus via infinitesimals is obviously very natural, and mathematicians (e.g. the pioneers of calculus, Euler etc...) have used arguments using infinitesimals long before they should have been really allowed to do so. Keisler's book (and in general the area of `Non-standard analysis') makes rigorous our intuition regarding infinitesimals, and is a set of rules that teach us how to formally reason with them. In my opinion this system is intuitive, but the student can never really have a proper understanding of what they are doing "from the ground up" with out some basic knowledge of model theory. The limit approach is less intuitive, but at least a student doesn't have to just accept some rules without truly understanding what's behind them. Possibly this infinitesimal approach is a half way house between teaching it properly with limits and just teaching rules of differentiation to people who aren't interested.

Answer 27

As an alternative to the definition of the concept of derivative by using limits, there is also the definition used in a book title Calculus Unlimited.

The value of the derivative $f'(a)$ is $\ge m$ if (but not only if) there is some open interval about $a$ within which $f(x) \left\{ \begin{array}{c} > \\[4pt] < \end{array} \right\} f(a) + m(x-a)$ according as $x \left\{ \begin{array}{c} > \\[4pt] < \end{array} \right\}a$.

But as for using any definition to find $\dfrac{d}{dx} x^3$, one could simply omit that nonsense and give them problems like this:

$$ (fg)'(x) \overset A = \lim_{w\to x}\frac{f(w)g(w)-f(x)g(x)}{w-x} \overset B = \lim_{w\to x}\frac{\overbrace{f(w)g(w) - f(w)g(x)} + \overbrace{f(w)g(x) - f(x)g(x)}}{w-x} $$

$$ \overset C =\lim_{w\to x} \left(f(w)\frac{g(w)- g(x)}{w-x} + g(x)\frac{f(w)-f(x)}{w-x} \right) $$

$$ \overset D =\left( \lim_{w\to x} f(w)\right) \left(\lim_{w\to x} \frac{g(w)-g(x)}{w-x}\right) + \left(\lim_{w\to x} g(x)\right)\left(\lim_{w\to x} \frac{f(w)-f(x)}{w-x}\right) $$

$$ \overset E = f(x)g'(x) + g(x) f'(x). $$

(a) What statement is proved by the argument above?

(b) One of the steps labeled $A$ through $E$ above uses the definition of "derivative" twice? Identify it and explain your choice.

(c) One of the steps uses the definition of "derivative" just once. Identify it and explain your choice.

(d) Two of the steps use only algebra and require no knowledge of limits. Identify them and explain.

(e) One of the steps uses properties of limits discussed in Chapter 2. Identify it and explain.

(f) One of the steps uses the fact that differentiable functions are continuous. Identify it and explain.

Answer 28

There was a recent article in the American Math Monthly, Analysis with Ultrasmall Numbers, that might be of interest. For a summary of its implementation in a high school classroom, see http://maths.york.ac.uk/www/sites/default/files/odonovan-slides.pdf.

A quick skim of its implementation seems to suggest that it provides a groundwork for some of the informal manipulations used in calculus-based physics classes.

Answer 29

Since this was recently bumped up to the top of the list, I would challenge the basic assumption expressed in the title of the question. In fact we don't all teach calculus using limits; I teach it using infinitesimals. The basic ingredient that replaces epsilon-delta limits in this approach is the shadow relating an infinitesimal-enriched continuum and an Archimedean continuum. Once students understand the basic notions of the calculus such as continuity and derivatives, we present the epsilon-delta paraphrases of the infinitesimal definitions.

Answer 30

I'm interested in the differentials-based approach advocated by Dray and Manogue at Bridging the Vector Calculus Gap. This is for multivariable calculus, but they do discuss the one-variable version (pdf). As they mention, people have reviewed calculus (especially for science courses) in these terms, but has anybody lately taught it this way?

(Also, the theory behind this approach is a little unclear beyond the first derivative, which is what led me to this question.)