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Here's how I see it (please read the following if you can, because I address a lot of arguments people have already made):

Let's take instantaneous speed, for example. If it's truly instantaneous, then there is no change in $x$ (time), since there's no time interval.

Thus, in $\frac{f(x+h) - f(x)}{h}$, $h$ should actually be zero (not arbitrarily close to zero, since that would still be an interval) and therefore instantaneous speed is undefined.

If "instantaneous" is just a figure of speech for "very very very small", then I have two problems with it:

Firstly, well it's not instantaneous at all in the sense of "at a single moment".

Secondly, how is "very very very small" conceptually different from "small"? What's really the difference between considering $1$ second and $10^{-200}$ of a second?

I've heard some people talk about "infinitely small" quantities. This doesn't make any sense to me. In this case, what's the process by which a number goes from "not infinitely small" to "ok, now you're infinitely small"? Where's the dividing line in degree of smallness beyond which a number is infinitely small?

I understand $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ as the limit of an infinite sequence of ratios, I have no problem with that.

But I thought the point of a limit and infinity in general, is that you never get there. For example, when people say "the sum of an infinite geometric series", they really mean "the limit", since you can't possibly add infinitely many terms in the arithmetic sense of the word.

So again in this case, since you never get to the limit, $h$ is always some interval, and therefore the rate is not "instantaneous". Same problem with integrals actually; how do you add up infinitely many terms? Saying you can add up an infinity or terms implies that infinity is a fixed number.

jeremy radcliff
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    In a sense I agree, in the classical framework the instantaneous rate of change is not really a "rate" which is maintained for any quantity of time, which is somehow problematic. At the same time, we are allowing for arbitrary resolution by working this way. If we don't do this, then now all functions are piecewise linear, possibly with extremely small pieces. But then how do you pick how small the pieces should be? – Ian Jun 11 '15 at 19:11
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    As an aside, an absolutely convergent series really is a proper generalization of a "sum", because you can shuffle the terms all around and still get the same result. This isn't built into the definition, but it shows us that at least in this setting our definition makes sense. – Ian Jun 11 '15 at 19:11
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    A similar question was posed a while ago. – pjs36 Jun 11 '15 at 19:13
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    Sum of a infinite series means that as I keep on taking the partial sums the values keeps approaching a certain value –  Jun 11 '15 at 19:14
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    This isn't the main focus of your question, but as someone with an affinity for infinite series I'll add my two cents: Saying that you "add up infinitely many terms" does not mean add infinitely many things and then stop. As you say, that would imply infinity has an end, and must be finite. Instead it defines a process that never stops. For example the sum $$\sum_{n=0}^\infty 2^{-n}$$ is shorthand for $$\lim_{N \to \infty}\sum_{n=0}^N 2^{-n}$$ Some of these processes converge. We say the limit of that sum is $2$, because we get arbitrarily close to $2$. $N$ can be as big as we want. – graydad Jun 11 '15 at 19:16
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    Instantaneous speed: Are you talking of Heaviside like jumps of some quantity, here position, and its derivative the Dirac delta as rate of change? – mvw Jun 11 '15 at 19:19
  • @Ian, re: "how small should the pieces be?". Couldn't you say "as small as you want them to be"? Which would still be different from claiming that there is such a thing a true continuity. I've always though of "continuous" functions in this way anyway. We allow for infinite granularity as a potential because we want to be able to input any value into the function, but ultimately it is still a discrete process. Is it wrong to think of it this way? – jeremy radcliff Jun 11 '15 at 19:28
  • @Ian , re: sum. Can you really say you're shuffling the entire sum, though? I think you can shuffle the terms of a partial sum, but shuffling infinitely many terms still doesn't make sense. – jeremy radcliff Jun 11 '15 at 19:30
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    It isn't discrete. Maybe the physical process you're trying to model actually is discrete somehow, but the mathematical object we've constructed is truly continuous. (As an aside, the question of whether physical space is quantized is open; to my understanding this question hinges on how quantum gravity works.) – Ian Jun 11 '15 at 20:01
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    As for the issue with absolutely convergent sums, what I mean is that if an infinite sum is absolutely convergent, then any sum consisting of the same set (multiset, perhaps, if some terms are repeated) of values converges to the same value. This is easy to state formally: if $p$ is a bijection from $\mathbb{N}$ to $\mathbb{N}$ and $\sum_{n=1}^\infty a_n$ converges absolutely, then $\sum_{n=1}^\infty a_n = \sum_{n=1}^\infty a_{p(n)}$. (This is not that easy to prove, though.) – Ian Jun 11 '15 at 20:03
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    I also think that you are artificially limiting your understanding by thinking of limits and infinity as "if you could ever get there then you would get this". Studying some (elementary) set theory might help with this. – Ian Jun 11 '15 at 20:08
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    If you are not yet familiar with the paradoxes of Zeno, you should read up on them; your concerns were first expressed millennia ago, and there is a fair amount of literature discussing these seeming paradoxes. – Eric Lippert Jun 11 '15 at 20:19
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    Do you have a similar problem with instantaneous position? If not, then what is the relevant difference between instantaneous velocity and instantaneous position of a moving object? – Eric Lippert Jun 11 '15 at 20:21
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    @EricLippert, The conclusion is probably that I'm not actually clear on what position is, but in my mind position can be described independently of time, whereas velocity is defined as the ratio of position to time, whereby if time is 0 (an instant in time, as opposed to an interval), then you have position/0, which makes no sense. I guess I just see velocity as having to be defined over a time interval. – jeremy radcliff Jun 12 '15 at 00:45
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    Do you understand the epsilon-delta definition of limits? – hobbs Jun 12 '15 at 00:47
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    In physics, at least, something is "instantaneous" if it's small enough to be ignored. Or too small to be measured. (Though I don't know if physicists call it that.) – Akiva Weinberger Jun 12 '15 at 01:04
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    I think of it this way... Imagine you are driving a car, down a highway. At some point, you take a picture of the view out of the windshield, your dashboard, and all the meters. Surely, this picture represents a single moment, an interval of time with zero length (it's just a picture after all), and, your speedometer will have a reading, in that picture, that captured instant. – Jonathan Hebert Jun 12 '15 at 02:07
  • @hobbs, i think I do. I think of epsilon as "give me an interval as small as you want and by continuing to add terms to my infinite series, I'll always be able to get closer to the limit than the epsilon you defined". Please let me know if this is wrong. – jeremy radcliff Jun 12 '15 at 02:27
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    @jeremyradcliff I wouldn't include the "infinite series" bit, we're talking about the behavior of the real function. But otherwise it's good. The fact that you can take the limit down to zero and approach some value for the rate of change justifies calling it an "instantaneous" one. – hobbs Jun 12 '15 at 02:32
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    @Rememberme: that's just one of many ways in which you could define what the sum of a series means; and, as the problem with conditionally convergent series shows, not a very good one. – Martin Argerami Jun 12 '15 at 04:49
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    Regarding "infinitely small" quantities: in normal calculus, these are just limits: quantities that approach zero. but you should check out Non-Standard Analysis, which treats infinitesimals as actual numbers (not just a concept but actual, rigorously defined objects). Infinitesimals are "hyperreal" numbers that are smaller than every real number but still not equal to zero. This takes away the idea of a "moving limit" while still providing an explanation for "instantaneous" rates of change. – chharvey Jun 12 '15 at 04:55
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    Maybe you should try not thinking of limits as movement, because then, as you say, you never "get there". Rather, when you see a limit like the derivative, imagine that there is a number that you cannot calculate, but that you can approximate with arbitrarily high precision. This is not some fuzzy thinking that cheats by evading the concept of instantaneous rate of change; if you can approximate a number arbitrarily well, then you know exactly what it is, even if you cannot calculate it "directly". (...) – Javier Jun 12 '15 at 17:51
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    (...) With this point of view, the limit is not a process that will never end. Instead, it's an indirect way of specifying (without ambiguity) a number that you couldn't otherwise calculate. Maybe this will help. – Javier Jun 12 '15 at 17:51
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    To reword what others have said, 'instantaneous' is just a metaphor to help with intuition. In the metaphor, there's a contradiction, movement over no time at all which really isn't movement. So that's a problem with the intuition, not the formal mathematics. – Mitch Jun 12 '15 at 18:25
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    You can see The Analyst and the complete text. – Mauro ALLEGRANZA Jun 12 '15 at 19:58
  • ...and this is why we (should) teach what real numbers are before calculus. – guest Jun 14 '15 at 07:23
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    A bit late, but I think of this as "small enough", not "instantaneous". When $h$ is small enough, the change gets as close are you like to the desired value. – simonzack Jun 14 '15 at 14:43
  • A Primer of Infinitesimal Analysis, John L Bell. –  Aug 19 '15 at 06:28
  • I see it this way: The instantaneous rate of change at a point gives you a measure of that point's contribution to (or role in) the change of the values of the function, describing to you how the function changes one point before and/or after that specific point. It tells you if you could continue your journey one unit further or one unit back under the same circumstance governing that point, how much change you would experience. So, a tangent line can be seen as a collection of points which have contributions to the change of the values equal to that point on the graph of the function. – Kaveh Rad Jul 26 '22 at 16:36
  • I’m voting to close this question because it has already attracted a fair amount of low quality answers that had to be deleted. – Kurt G. Aug 15 '23 at 07:56

14 Answers14

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In math, there's intuition and there's rigor. Saying $$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h$$ is a rigorous statement. It's very formal. Saying "the derivative is the instantaneous rate of change" is intuitive. It has no formal meaning whatsovever. Many people find it helpful for informing their gut feelings about derivatives.

(Edit I should not understate the importance of gut feelings. You'll need to trust your gut if you ever want to prove hard things.)

That being said, here's no reason why you should find it helpful. If it's too fluffy to be useful for you that's fine. But you'll need some intuition on what derivatives are supposed to be describing. I like to think of it as "if I squinted my eyes so hard that $f$ became linear near some point, then $f$ would look like $f'$ near that point." Find something that works for you.

Parcly Taxel
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Zach Stone
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    The best I've come up with thinking about the limit definition of a derivative, is "if there were such a thing as an instantaneous rate of change, this is what it would be", kind of the way you could think of a limit as "if we could ever get to infinity, that's the value we'd get". Is this wrong in the sense that I could limit my understanding of certain math concepts in the future by adopting this view? – jeremy radcliff Jun 11 '15 at 19:46
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    It seems to me that you understand what's going on. Limits are weird at first, and derivatives are usually people's first exposure to serious limits. If you keep asking good questions, I suspect you'll be just fine. – Zach Stone Jun 11 '15 at 20:20
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    This answer seems to neglect the fact you can use infinitesimals to build a rigorous definition of calculus. – Yakk Jun 12 '15 at 19:33
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    @Yakk: Not very well. Non-standard analysis requires you to keep shifting back and forth between working in the hyperreals and restricting your attention to standard reals, and the only principle for when to do what seems to be "think back to your knowledge of ordinary limit-based analysis and fudge your choice so you get the result you already know is right". – hmakholm left over Monica Jun 13 '15 at 12:21
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    @Henning NSA doesn't require you to shift back and forth between reals and hyperreals, for example there are no hyperreals in the IST approach. – Matthew Towers Jul 11 '15 at 13:36
  • @mt_: The words may be different than "hyperreals", but IST certainly has a distinction between standard and non-standard reals. – hmakholm left over Monica Jul 11 '15 at 16:23
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    Interesting remark "You'll need to trust your gut if you ever want to prove hard things." I've been told by some people that at some point, things may become so unintuitive and abstract that you can't rely on your feelings to prove things anymore. – Ovi Oct 25 '16 at 05:36
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    The snarky response is to "have better feelings." The serious response is that much of the hard work of mathematics is to build your intuition even for highly abstract fields. Forcing, for example, is outrageously abstract (compared to arithmetic, anyway). But it's useful to feel like forcing is 'adding the right member to a model.' The rigor is abstract, but it's practical to have intuition with simple terms. On the other hand, some intuition typically becomes less helpful at high level. E.G, 2D geometry intuition sucks in Banach Spaces. So we must find better, more helpful feelings. – Zach Stone Oct 25 '16 at 18:55
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The idea behind $$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ is the slope of the graph $y=f(x)$ .

Now for a moment forget about your instantaneous velocity and think about your average velocity. What is average velocity? I think average velocity is $$\frac{x_f-x_i}{t_f-t_i}$$. Now if look at this carefully this is my slope but in a given interval of time.

So you might question what is the difference between instantaneous velocity and average velocity , Both of them talk about intervals .

No that not the idea over here. Now if it had been a linear graph it would have been very easy to calculate your instantaneous velocity , but in your you graph that's not the case. Here you see the particle (or object) is changing its velocity every moment of time and that becomes impossible to deal with . So to remove this element of doubt what we do is try to take the limit and try to get close to the frame of time and try finding the velocity and name it instantaneous velocity.

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Instantaneous Rate of Change at a Point, not over an Interval

You can describe rate of change without any interval at all, at a specific point. In fact, to try and involve an interval, even a very small one, is somewhat counter to the concept. If you involve an interval in describing a rate of change, then you would be talking about an average rate of change. For example, when people describe the velocity(rate of position change) of a vehicle over an interval of 100ft, then you would usually be talking about the average speed, assuming it wasn't constant over the entire interval.

Therefore, a rate of change is always instantaneous. You can identify any point, which has no length/interval, and say what the rate of change at that point is. Instantaneous rate of change is analogous to a point.

This also helps understand why certain points have an undefined rate of change, and why the "ever smaller interval" concept is somewhat misleading.

enter image description here

If you tried to use the very very tiny interval concept to identify the rate of change across point p and point q by sampling a point just to the left and right of p to define an interval, then you'd see a very large change. Perhaps point p is at x coordinate 1.00, and we define an interval from .99 to 1.01 and sample values of .99,2 and 1.01,10. This would be a rate of change of 400 y per x. If you make your interval smaller and smaller, your rate of change appears to approach infinity, which again is very misleading, because the instantaneous rate of change at p is undefined.

I'm not trying to disregard the intervals concept as useless, but to emphasize that very small intervals are a concept that can't be applied rigorously. They certainly help understand the concept of a derivatives, because if you were to imagine that you don't know what the instantaneous rate of change at a particular point is, then you'd have to observe how much change occurred over an interval in order to estimate the rate of change. It helps to understand how these functions were "designed".

There is no observed change, but we know the RATE of change

Let's take instantaneous speed, for example. If it's truly instantaneous, then there is no change in x (time), since there's no time interval'

You are absolutely right! There is no change in position as there is no defined interval. However, at a specific point, we can still describe what the instantaneous rate of change is, which is a way of saying how much it will change if an interval were defined. Imagine we freeze a frame of a movie where someone tells you a bullet just exited a barrel at an instantaneous velocity of 300 meters/s. That's not hard to understand right? If someone asked, disregarding air resistance and gravity, where will that bullet be two seconds from then? Now you've introduced an interval, and using two things, the initial position, and the instantaneous rate of change at that position, you can confidently draw a line along the bullet's trajectory and conclude it will be 600 meters away in that two seconds. Based on that instantaneous information, we can model for any interval how that bullet's position will change. We can apply the instantaneous velocity across the entire interval only because we assumed it was constant across the interval.

Where calculus comes to the rescue is that we often deal with non-constant rates of change. We know in reality that the bullet will slow down due to air resistance A(), and it's trajectory will be an arc instead of straight line due to the constant acceleration of gravity G() (which results in a non-constant velocity). If we have functions that describe air resistance and gravity's influence on our velocity, then we can describe the non-constant instantaneous velocities over any point on the entire interval. Since those are instantaneous points and are non-constant, they are somewhat useless. We could use the "very small interval" approach of sampling changes in position to sum up across the entire interval, but since the velocity is constantly changing, we'd be accumulating alot of error. E.g. if the initial velocity is 300 m/s but slows to 250 m/s after half a second, then how far did the bullet travel in half a second? You could try to average, but that assumes linear change in speed which might not be the case. Just about any iterative approach you take that involves small intervals will introduce error. However, with the help of calculus we can directly and accurately incorporate the influence of A() and G() on our non-constant velocity to directly calculate the change in position.

More than just a Concept

In addition to being an aid in understanding concepts, the "very small interval" concept is often applied in computational models where direct methods are not feasible, perhaps because a computer algorithm cannot determine a direct method. This is exactly how physics are simulated in many computer games. Each iteration it may know that .03 seconds has passed, and based on various rates of changes(usually determined based first on what forces are in affect) it will determine what has changed across that interval. In certain situations this can be extremely inaccurate, but in other cases with small enough intervals it may be sufficiently accurate to meet desired requirements.
AaronLS
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    I'm rereading the answers to my question after several months of Calc (and a lot more thinking) and your answer is very helpful and makes a lot more sense than it did then, thank you! – jeremy radcliff Sep 27 '15 at 02:45
  • @jeremyradcliff This is a better Answer than the currently-accepted one, which I disagree with. – ryang Oct 07 '21 at 19:44
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Think of a mountain: far away it's nearly flat; near the top it's more steep.

If you ask "How steep is it where I am standing right now?", you're asking exactly the same thing as "What is the rate of change of my elevation with respect to my position right here?"

I feel that should be pretty intuitive; that's what instantaneous rate of change means.

(Side note: in this case, you can move in two independent directions: north/south and east/west. That means there is a separate rate of change of elevation for each of those directions at every possible position on the ground. But in your case, you're only dealing with time, which can only go forward/backward, and hence there's only one number to worry about at each instant, not two.)

user541686
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As the accepted answer stated, the English language statement is informal and exists to provide intuition only.

That does not mean it might not be finessed.

If one rewords to 

Rate of change applying at a specific instant

then I think this is consistent with:

  1. A rate of change being defined only on intervals
  2. Those intervals being arbitrarily small
  3. We have a unique value meaningful for a given instant, provided of course that the limit exists.
Keith
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I will answer your question with a question.

You are driving a car. 60 miles per hour is 88 feet per second (which is pretty cool in and of itself. You look down at a text for 1/2 second? You moved 44 feet.) Now, when I ask where you were in a given second, you could have been anywhere within +/- 44 feet. So you say "which 1/100th second are you asking about?" But 44 feet is 528 inches, so even if I say at time = 6.31 seconds past 2:30, you only know within a 10 inch range.

Can you ever tell me precisely where you are if you are in motion?

Instantaneous rate is a calculus concept, but it should be understandable to you as the slope of a tangent line.

enter image description here

The line tangent to the parabola has a slope of 2. Even though the "slope" is just a point.

JTP - Apologise to Monica
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I think the answer of @ZachStone already expresses the essence of your question.

I would add a more Physical/Engineering point of view, because you call in the intuition, and in applied maths, as usually found in engineering, you need a lot of intuition.

As a concrete example, let's consider the speed of a car (or of any point mass). There is the average speed which is trivially and very understandably defined as:

\[ v_m = \dfrac{\Delta s} {\Delta t} \]

You define the instantaneous speed as the limit of that ratio as the time interval tends to zero:

\[ v = \lim_{\Delta t \to 0} \dfrac{\Delta s} {\Delta t} \]

Of course if you knew a mathematical expression for s(t) you could do all your usual math, and that's ok. But what does it mean that limit in practical applications? It means that you look for a value of the "average speed" over a "very small" time interval, where "very small" means "as small as possible in relation to the application". That usually means that $\Delta t$ is so small that s can be deemed constant over that time interval, because the allowable measurement error is larger that the variation of s over that interval.

In a sense, you approximate your motion as you go ahead: at any time instant you say to yourself "if I had to approximate now the motion with a constant speed motion, which speed would the car have?". The answer is "the instantaneous speed" (which is, as I said before, approximately computed as an average speed over a "small enough" time interval).

LorenzoDonati4Ukraine-OnStrike
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I think this question should belong into meta, as it is a question between finitism and platonism, but I try to answer you as best as I can:

The dividing line where a number $\epsilon$ is small enough is when someone gives you a number $\delta>|x-c|>0$, there exists $\epsilon>|f(x)-L|$, where $c$ is a point near $x$, $L$ is the limit. So you need not think $h$ as an interval, but as a number.

In arithmetic sense, you can add infinitely many terms if they follow a rule which can be mathematically constructed: the geometric series is a good example. The existence of integral is therefore quite natural.

Addressing your first question in a philosophical manner:

Say that there exists events that happen instantaneously. If they do exist, they need to be observed. For the observation to happen, time needs to pass, so we can think "instantaneous" as an event which occurs in the least time possible for the event to actually happen.

Eemil Wallin
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    You misunderstand the purpose of meta forum - it is for discussing the website Math Stack Exchange and its operation. Mathematical questions of any sort are off-topic there. – Zev Chonoles Jun 11 '15 at 19:24
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This was stated in another form in a different answer, but I thought I'd say it in a way that might be conceptually easier to grasp:

The instantaneous rate of change at any point in a function is the slope of the line that is tangent to the function at that point.

If you imagine a line that intersects two points on an arbitrary function that are some arbitrary distance apart, and then slide those points together, you will see that the slope of the line changes depending on the distance between them. When the two points are so close together that they become one, then the slope of the line is equal to the instantaneous rate of change of that function.

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thomij
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Why don't we just say "approximately instant" to appease our intuition, and then define our velocity as an arbitrarily close approximation to an average one?

As a bit of an aside, 'closeness' is tied to so many higher levels of Mathematics. So many things hang on it. Topology, Analysis, and the like to be sure. So, as mathematicians, we've become comfortable with the idea of the limit as a definite value to make so many other things work out.

I always have in the back if my mind that that denominator IS NOT zero but gets really close to it. And for me the limit is hanging onto that closeness and rejecting the idea that it can actually be zero - because it can't be.

John Molokach
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    I should add here that epsilon and delta are precisely the differences between the quantity that is approaching zero and zero itself. If you study higher level mathematics, you will use epsilons and deltas many times over! – John Molokach Jun 12 '15 at 12:58
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The expression $\dfrac{x^3-8}{x-2}$ is undefined at $x = 2$. When $x = 2$ you get $\dfrac 0 0$, which is undefined. So suppose we decide to define the value of $\dfrac{x^3-8}{x-2}$ when $x = 2$. What number makes the most sense and why? Well, when $x \ne 2$ we can simplify

$$\dfrac{x^3-8}{x-2} = \dfrac{(x-2)(x^2 + 2x + 4)}{x-2}=x^2 + 2x + 4$$

and $x^2 + 2x + 4 = 12$ when $x = 2$.

If you graphed $y = \dfrac{x^3-8}{x-2}$ you would get the graph of $y = x^2 + 2x + 4$. Depending on how good your graphics software is, it may or may not indicate that there is something wrong at $x = 2$, but the point $(2, 12)$ is actually missing. We call points like this "removable discontinuities".

A rate of change of a function $f(t)$ between a variable time $t$ and a fixed time $a$ looks like $\dfrac{f(t) - f(a)}{t-a}$ and you notice that, if you let $t=a$, you get $\dfrac 0 0$.

A whole lot of the time, $ y = \dfrac{f(t) - f(a)}{t-a}$ has a removable discontinuity at $t = a$. If the missing point is (a, A), then it turns out that $\displaystyle lim_{t \to a}\frac{f(t)-f(a)}{t - a} = A$. The number A is called "the derivative of $f(t)$ at $t = a$" and it is described as "the instantaneous rate of change of $f(t)$ with respect to $t$ at $t=a$.

Steven Alexis Gregory
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What makes Calculus "useful" in the "real world" isn't the formalism about limits or infinitesmials1. It is the fact that most functions in the "real world" are relatively smooth and well behaved.

Such functions act linearly if you zoom in "enough" -- they act like little linear functions (well, affine functions -- linear with an offset).

The derivative associates each point on a function's domain (valid input) with a linear function. This describes the "slope" at that point.

If you are "close enough" to the point, then the linear function that the derivative gives you is a good approximation to the original function.

As linear functions are ridiculously easier to work with than other functions, this makes understanding the original function easier. Even if we are forced to understand an infinite number of such linear functions to understand a single non-linear function (the family of derivatives).

If a function goes from $\mathbb{R}$ to $\mathbb{R}$, then the linear maps have a natural bijection with elements of $\mathbb{R}$ -- so instead of the derivative returning a function, we denote it as returning a single real value.

If $f'(x) = g(x)$, then the affine map that approximates $f$ near $x_0$ is $h(x) = f(x_0) + g(x_0)*(x-x_0)$. Most of that is boring, so we just talk about $g(x_0)$.

In higher dimensions, the value returned from "the derivative" is a higher dimension linear function, which is represented by a matrix.

These linear functions are "guaranteed" to be "useful" if you are close enough to the point in question. How close can vary with the function in question (see the error bounds on prefixes of the Taylor series).

This is the "slope at the point" because talking about the interval around the point where it is half-decent is hard to talk about, but we do know that if we make the area small enough it is going to be good.

1 There are multiple formulation of calculus that acts on infinitesmials -- the smooth analytic analysis or non-standard analysis.

In it, you actually talk about points $x+h$ infinitesimally close to $x$, and evaluate $\frac{f(x+h)-f(x)}{(x+h)-x}$. The result is the derivative of $f$.

The delta -- $h$ -- is smaller than every classical real number. In a sense, $x+h$ is "still at $x$" (as far as classical real numbers are concerned), but in the non-standard analysis it isn't the same value.

Under this formalism, "at the point" could refer to "at the point, and infinitesimally close to the point".

Yakk
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    $(f(x+h) - f(x)) / ((x+h) - h)$ is not the derivative of $f$: it is merely infinitesimally close to the derivative (assuming the derivative exists). –  Jun 13 '15 at 06:39
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To talk about instantaneous rate of change, you have to first have a sufficiently nontrivial idea of an instant. The main point of differential calculus is to flesh out the idea of an instant, how to do calculations in instants, and how to use integrals to "add" them all up.

In a more precise sense, the central idea is that at a point, you can have numbers that encode what happens near that point.

The simplest example of this is the meaning of continuity: if you know $f(a) = b$, then you also know that $f(c) \approx b$ whenever $c \approx a$. So when you're working with continuous functions, knowing what happens at a particular point means you have some idea about what happens near that point too. Another example of something you know is that $\lim_{x \to a} f(x) = b$.

The derivative is another example of this, and there is a lot of circumstantial evidence that links it to being a rate of change. For example:

  • The mean value theorem, differential approximation, and similar things allow you to use a derivative to estimate a rate of change. e.g. $$y - y_0 \approx (x - x_0) \frac{\mathrm{d}y}{\mathrm{d}y}$$
  • I can use an integral to "accumulate" the derivatives to produce an actual change: $$\int_a^b f'(x) \mathrm{d}x = f(b) - f(a)$$
  • If you plot the graph of the rate of change (i.e. the difference quotient) of a differentiable function between $x_0$ and $x$, there is hole that really ought not be there (in a very similar sense how one would like to say that $t/t = 1$). The derivative is the value that fills that hole.
  • In many circumstances (all of them, when suitably treated), one can compute a derivative formally by the procedure of writing down $(f(x) - f(x_0))/(x - x_0)$, doing simplifications to cancel out the denominator, then making the substitution $x \to x_0$.

Rather than being stuck on the fact that pre-calculus tools fail to express the idea of a rate of change in the situation where the derivative applies, you should instead be looking to see how the method of using derivatives allows you to study the idea of rates of change in situations where the pre-calculus tools were unable to help.

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I had come to accept the slope intuition of derivatives a long time ago, when I first learned calculus. I never really understood it completely, I must confess, but since it proved useful in so many different applications (particularly in physics), I began to accept that interpretation without having to think about it at all after a certain point.

It was when I studied calculus slightly more formally that I gained a better understanding of the concept of limit and derivatives as a whole. The breakthrough came when I understood how sequences/series of numbers and their limit points work. Consider the following infinite sum:

\begin{equation} 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... \end{equation}

The 'limit' of this sum is equal to $2$. Now obviously, you can't go on adding infinite terms to get exactly $2$, yet we say (and we say this as a proper mathematical statement - not as an ambiguous expression of out intuition) that the limit of the sequence is $2$.

The idea behind the limit (in the case of sequences/series) is that if you kept on evaluating the sequence with more and more terms, you would come closer and closer to $2$. You wouldn't be 'converging' to $\textbf{any other number}$ other than $2$. We call $2$ a limit point of the sequence.

In a similar manner, when we talk about 'instantaneous' rates, what we are saying is that if we calculate the rate with smaller and smaller intervals, we get closer and closer to the value we call the 'instantaneous' rate. We never actually attain that value, but we get closer and closer to it.

In a sense, it's kind of like a 'sacred' value that your expression is always striving to reach. You can't ever really be sure of it even if you study the behaviour of the expression for very long. As an example, you might conclude that the sequence I showed above has a limit of $2.000000000001$, just by computing the value with more and more terms. But if you went even further, you would discover that the sum's distance from $2.000000000001$ begins to increase after some time (suppose you hit the midpoint of $2.000000000001$ and $2$, the true limit; continuing with the evaluation of the sum from there on out would lead you closer to $2$, and thus farther away from $2.000000000001$).

So you see, the thing about the limiting value of an expression is that it is not an easy task to determine it just by studying the way the expression behaves numerically (which is what the intuitive explanation of 'instantaneous' may lead you to believe). This is why the mathematics of limits is so profound. You can compute the $\textbf{exact}$ value that the expression converges to, regardless of 'practical artefacts' like numeric precision. It is actually quite an elusive value, the limit of an expression...but it has a lot of meaning to it.

If you continued forever in the direction of the limit, you would continuously get closer to the limiting value (limit point), but you would never reach it. If we could measure the rate in an intervals of ever decreasing length, we would approach the instantaneous value - but we never get an exact match.

Rishabh Malviya
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  • I was hoping somebody already gave an answer like this so I wouldn't have to. If we compute average velocity on smaller and smaller intervals and observe a sequence like $2.01$, $2.001$, $2.0001$, $\dots$, then we appear to be converging to the "true" velocity of $2$ at that instant. – Austin Mohr Dec 04 '16 at 03:35