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

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.

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.

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.