There are certainly other rigorous approaches to the derivative out there. 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. It is worth noting that there is a lot of historical precedent for teaching it as a limit, which occurs already in Euclid. 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.
Strictly, he says equivalently that one cannot interpose another line between the tangent and the circle itself, i. 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.
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.
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. However, I recommend you teach it any way that makes sense to you.
Make up your mind what seems important to you, and go for it! 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. 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.
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.
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. 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:. 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.
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.
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. Identify it and explain your choice. Identify them and explain. Identify it and explain. There was a recent article in the American Math Monthly, Analysis with Ultrasmall Numbers , that might be of interest.
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. 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. 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.
Sign up to join this community. The best answers are voted up and rise to the top. Why do we teach calculus students the derivative as a limit? Ask Question. Asked 11 years, 1 month ago. Active 3 months ago. Viewed 79k times. Improve this question. But what would be the point of teaching students the symbolic rules as axioms without explaining to them how they are derived? Would you advocate teaching maths undergraduates the combinatorial properties satisfied by character tables of finite groups, so that they can work out the tables in most cases, without proving any of the properties or maybe even without explaining what a character is?
It is already common for students to have a black-box view of mathematics; I don't think you want to encourage it. Perhaps you want to begin with the definition via limits and then derive the rules from there. Emphasize to your students that "Why didn't we just use the rule from the start? The rule is a consequence of the definition, not a self-evident truth. That's simply the most effective definition of derivative the nonstandard analysis one would require a knowledge of logic that no freshman is supposed to have!
The geometric problem of computing tangent lines is natural and easy to motivate; the limit definition is reasonably easy to motivate from the geometric problem; and then students could spend reasonable amount of time flailing around trying to compute derivatives of different functions.
Show 20 more comments. Active Oldest Votes. Improve this answer. Deane Yang. But then again, it is a deceptively deep concept. As for modern treatments that emphasize other things than the standard, did you ever look at "Calculus in Context", the five colleges calculus? Available at math.
But that's the hard way. For an engineer or physicists, who thinks in units and dimensional analysis and views the derivative as a "sensitivity" as I've described above, the answer is dead obvious.
In fact, I advise being pragmatic and teaching in a fashion that will not alienate you from your department or school administration. That said, if you want to slip in more understanding which I claim actually helps students learn the symbolic methods better , I recommend taking problems from the Harvard calculus textbook, as well as their precalculus text "Functions Modeling Change".
Especially those where no formula is given for the function. That is, the box behavior should be single-valued. Otherwise, we might imagine a black-box that accepts a given input and outputs a random number, perhaps different every time, and although this accords with your description, it is not a function.
The derivatives are always computed numerically and under the hood. What your finance students need to know is how to interpret and use these numbers.
This is presented rather well in my view in the first few chapters of the Harvard Calculus text. Show 41 more comments. Jeff Strom. I do not see how these rules could ever access a function which is not a polynomial. Show 2 more comments.
Here's a quote from Picasso of all people on teaching: So how do you go about teaching them something new? Andrew Stacey. Of course, everyone who has ever taught has come across specific cases where it might be applied. I really liked the line because of several recent conversations about students who don't take notes, skip class, rush through homework, and don't ask questions.
Such students are the exception rather than the rule, but they can get under one's skin. For times like that, I thought Andrew's line would be a good substitute for the glib old saw about a horse and water. I think that Mark quoted enough context by quoting the second sentence.
I also disagree with your rephrasing because my original statement said, " If the students There's loads more to say on this, but MO is thankfully a lousy place to say it. So I will content myself with saying that I know Mark and I know that he takes teaching very seriously so I see his "cheerleading" of that sentence coming from the best possible motives.
Show 1 more comment. Pietro Majer. It has a lot of hypotheses that no-one ever checks, and students always apply it when the quotient is in the wrong form, so I won't teach it and you'd better not use it. I didn't mind when he said that because I was one of the ones who'd never heard of it.
De'LHopital himself would be embarassed to know somebody's still wasting time with such an awkward theorem like that thing that brings his name. Theorems, like cakes, don't always come out well; that thing came out very badly, and left a mess in the oven. Today, it may be at most of some historical interest. Teach the Landau notation instead! It's so close to the usual definition that I don't really believe that students have a significantly easier time with it.
However, I believe that when you teach calculus, this definition inspires you and you do a very good job teaching it, more so than you would with the standard definition. I suspect that most "the students find it easier when Viewed times. Turkhan Badalov. Turkhan Badalov Turkhan Badalov 1 1 gold badge 7 7 silver badges 16 16 bronze badges. Other examples include a lot of formulas like the binomial theorem and the quadratic formula. Or is there a name of the mathematician, who found them?
I mean explicitly shortcuts. Add a comment. Active Oldest Votes. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Like the product rule, there are a few different ways you might see the quotient rule represented. I recommend picking the one that makes the most sense to you so that you can memorize the formula. Unlike the product rule, the order does matter here. This is because we are using subtraction and division rather than addition and multiplication.
Since we need the pieces to be in the correct order, it is helpful to come up with some method for memorizing the quotient rule as I have shown it above.
We need to take the derivative of some function, that can be represented as a fraction made up of two functions that are easier to derive. And if you read it out loud, it almost seems to have a little jingle to it, which makes is easier to remember:.
Just like we did with the product rule example, we want to first recognize how we will split up this function. The main difference is that this distinction does matter with the quotient rule. Once we have made this distinction, we can consider these two functions individually for a moment and find each of their derivatives.
I already found these derivatives in Example 1 of The Product Rule. If you want to see this again, click here. So far we have:. Now that we have figured out these four parts, we can simply plug them into the quotient rule formula we have above.
This tells us that:. And you can see more quotient rule practice problems here. The product rule is a very useful tool to use in finding the derivative of a function that is simply the product of two simpler functions.
There are a few different ways you might see the product rule written. I would recommend picking whichever one is easiest for you to remember and understand so that you can work with it from memory. Although you may only need to remember one of these, you should be able to recognize the other representations as the product rule when you see them.
All of the following are different ways of writing the product rule:. As a result we can reorder the product rule equations shown above, so. As long as you multiply the first function with the derivative of the of the second and multiply the second function with the derivative of the first, then add the two together, it will work out.
The phrase I like to think about when I need to remember the product rule is:. I like that phrasing personally, but I recommend you come up with some trick for remembering the product rule because it comes up frequently. They have developed the power rule on their own!
The power rule is easily adopted by most students, as are the rules for constants, sums and differences, and constant multiples of functions. Combining rules requires students to write legibly a struggle for some! Some review of Algebra 2 concepts might be required today.
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