The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. We can formally define a derivative function as follows. Follow the same procedure here, but without having to multiply by the conjugate.
We use a variety of different notations to express the derivative of a function. To understand this notation better, recall that the derivative of a function at a point is the limit of the slopes of secant lines as the secant lines approach the tangent line.
We have already discussed how to graph a function, so given the equation of a function or the equation of a derivative function, we could graph it. The solution is shown in the following graph. First, we consider the relationship between differentiability and continuity. We will see that if a function is differentiable at a point, it must be continuous there; however, a function that is continuous at a point need not be differentiable at that point.
This illustrates that continuity at a point is no guarantee of differentiability -- the existence of a tangent -- at that point. Conversely, though, if a function is differentiable at a point -- if there is a tangent -- it will also be continuous there. The graph will be smooth and have no break. Since differential calculus is the study of derivatives, it is fundamentally concerned with functions that are differentiable at all values of their domains.
Such functions are called differentiable functions. To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" "Reload".
Think about this yourself first! Since the derivative is this limit : then the symbol for the limit itself is Read: "dee- y , dee- x. We are to take the derivative of what follows it.
The difference quotient is a version of. And at times we will use the latter. Hence the difference quotient is. That is,. The student should now do Problems that require the definition of the derivative. Please make a donation to keep TheMathPage online. E-mail: teacher themathpage. By the derivative of a function f x , we mean the following limit, if it exists: We call that limit the function f ' x -- " f -prime of x " -- and when that limit exists, we say that f itself is differentiable at x , and that f has a derivative.
This is what we wanted to prove. This symbol by itself: d dx "dee, dee- x " , is called the. 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.
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. 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. 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. 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:. 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. 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. First of all, this example is important in differential geometry e. 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. 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. 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? 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. 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. 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. So far no one's mentioned or did I miss it? They find all sorts of creative ways of getting things wrong when doing this. 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. 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?
Ok, now I will mention some personal opinions about teaching calculus. 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. Well, the definition of derivative is probably one of the best application of the notion of limit, from a didactical point of view.
I think this is beautiful and relatively simple, once you get the students to think about it for a minute. 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?
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.. This is to say that I think it is possible to have students learn this theoretical aspects of calculus, if high school kids do.
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. 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.
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? Many calculus students are adults but, ahem, need practice with inequalities. Gee, sensitivity coefficients are just derivatives, and they'll be estimated from the definition , not symbol-pushing.
It's funny that actually many students believe that the symbiosis is always the other way around, i. 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. 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. 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. 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 we had expressed this function in the form , we could have expressed the derivative as or. We could have conveyed the same information by writing. Thus, for the function , each of the following notations represents the derivative of :. In place of we may also use Use of the notation called Leibniz notation is quite common in engineering and physics.
To understand this notation better, recall that the derivative of a function at a point is the limit of the slopes of secant lines as the secant lines approach the tangent line.
The slopes of these secant lines are often expressed in the form where is the difference in the values corresponding to the difference in the values, which are expressed as Figure.
Thus the derivative, which can be thought of as the instantaneous rate of change of with respect to , is expressed as. We have already discussed how to graph a function, so given the equation of a function or the equation of a derivative function, we could graph it.
Given both, we would expect to see a correspondence between the graphs of these two functions, since gives the rate of change of a function or slope of the tangent line to. In Figure we found that for. If we graph these functions on the same axes, as in Figure , we can use the graphs to understand the relationship between these two functions.
First, we notice that is increasing over its entire domain, which means that the slopes of its tangent lines at all points are positive.
Consequently, we expect for all values of in its domain. Furthermore, as increases, the slopes of the tangent lines to are decreasing and we expect to see a corresponding decrease in. We also observe that is undefined and that , corresponding to a vertical tangent to at 0.
The graphs of these functions are shown in Figure. Observe that is decreasing for. For these same values of. For values of is increasing and.
Also, has a horizontal tangent at and. Use the following graph of to sketch a graph of. The solution is shown in the following graph. Observe that is increasing and on.
Also, is decreasing and on and on. Also note that has horizontal tangents at -2 and 3, and and. Sketch the graph of. On what interval is the graph of above the -axis? The graph of is positive where is increasing. First, we consider the relationship between differentiability and continuity. We will see that if a function is differentiable at a point, it must be continuous there; however, a function that is continuous at a point need not be differentiable at that point.
In fact, a function may be continuous at a point and fail to be differentiable at the point for one of several reasons. Let be a function and be in its domain. If is differentiable at , then is continuous at. If is differentiable at , then exists and. We want to show that is continuous at by showing that.
Therefore, since is defined and , we conclude that is continuous at. We have just proven that differentiability implies continuity, but now we consider whether continuity implies differentiability. To determine an answer to this question, we examine the function. This function is continuous everywhere; however, is undefined. This observation leads us to believe that continuity does not imply differentiability. For ,. See Figure. Consider the function :. Thus does not exist. A quick look at the graph of clarifies the situation.
The function has a vertical tangent line at 0 Figure. The function also has a derivative that exhibits interesting behavior at 0. We see that.
This limit does not exist, essentially because the slopes of the secant lines continuously change direction as they approach zero Figure.
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