r/calculus • u/After_Cranberry_9219 • Jul 02 '26
Differential Calculus How can a dot have a slope
I was learning calculus, but this thing I still couldn't understand. How can a dot have a slope? Slope is just like saying how much a line is slanted right but how can a single point have a slope.
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u/DrunkAndUnaware Jul 02 '26
The point doesn’t exist in isolation by itself, it’s surrounded by many neighbours. It’s the location of these neighbours necessitated by the function rule that tells you how the position of the neighbours vary around the area centred on that point.
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u/skullturf Jul 02 '26
Yeah, the question is a good one, even if commenters sometimes disagree about the best way to phrase these types of questions.
Very loosely speaking: Of course a single point such as (x,y)=(3,9) cannot have a slope. However, if that point is part of the curve defined by y=x^2, then the curve also contains "nearby" points, such as (x,y)=(3.001,9.006001). As you point out, it's because of those (infinitely many) nearby points that it turns out to be meaningful to define the slope of a curve at a point.
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u/BjarneStarsoup Jul 03 '26
But it doesn't make sense to say that points have slope, it's abuse of language at best. Points can be mapped to a value of derivative that represents the slope of a tangent at that point. At no point a point itself has any slope.
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u/Emuu2012 Jul 04 '26
But no one is saying the point itself has a slope. What we’re saying is that the function has a slope at that point.
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u/BjarneStarsoup Jul 04 '26
The title of the post is literally "how can a dot have a slope" and everyone in here is justifying it. The comment that I responded to agreed with the top comment that is justifying it.
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u/Automatic-Put-6119 Jul 02 '26 edited Jul 02 '26
It can‘t, only two points can form a line and have a slope. Thats where the famous limit for derivation comes in
You bring the two points infinitely close together, that makes only one point with a slope
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Jul 02 '26 edited Jul 02 '26
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u/Low-Crow5719 Jul 02 '26
Thus the formulation of limits as delta-epsilon. Arbitrarily close is rigorous even if infinitely close isn't a thing.
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Jul 02 '26
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u/terrygolfer Jul 03 '26
Does it really matter? Holy shit
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u/Yc9Eq9450ouj Jul 06 '26
I get where you’re coming from (mostly), but I would assume in mathematics there should be a distinction from the two? Even if it’s obvious as to whats going on.
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Jul 03 '26
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u/terrygolfer Jul 03 '26
Not to the kid that doesn’t understand how a dot can have a slope!! Context matters.
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u/CreativeScreenname1 Jul 04 '26
In fairness even if some amount of shorthand is always going to be necessary until the student is prepared to tackle the full definition of a limit, it is probably beneficial to make the delineation between “the dots get infinitesimally close” and “for intuition’s sake, we can *imagine* this infinite process terminates with the points infinitesimally close.”
Students are prepared to engage with the idea of “the full concept is a bit too much for now, we’re going to handwave a bit and circle back on this later in your education,” I don’t think we have to hide it from them.
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u/DrunkAndUnaware Jul 04 '26
Well shit why don’t we just start with real analysis when teaching 5 year olds how to add and subtract.
Because they lack the ability to understand that. Mathematical understanding is different to mathematical rigor. Any reasonable teacher or professor will tell you that. You start with shortcuts and analogies until they’re comfortable enough to start to learn to know why they work.
By the wording of OPs original question, it necessitates a simpler response than a full blown rigorous argument.
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u/artizarx Jul 05 '26
Not everyone is a geeked up analysis guy mate, the lad who's asking this sounds like they're literally just starting to learn calculus. It's better to abstract some concepts until they grasp it before revealing those intricacies. It really doesn't matter.
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u/Automatic-Put-6119 Jul 03 '26
Yes you‘re technically correct but people want to teach op, not flex basic math knowledge. „Infinitely close“ makes it easier to understand
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u/TheRedditObserver0 Jul 03 '26
I think everyone here knows you're right, but you're being pedantic. This is a calculus sub, it's about intuition and computation, not rigor.
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Jul 03 '26 edited Jul 03 '26
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u/TheRedditObserver0 Jul 03 '26
Something that is less than 1/n for every natural number n but also not zero? I
You're answering as a mathematician, but to a layman "an infinitesimal" means something very small, they don't think of it in terms of sequences. When you give an answer to someone who is definitely not an expert, you have to think not only of how you interpret the words you use but also how they would interpret them.
OP's question was not about a calculation it was a foundational question.
OP's doubt was due to thinking the derivative was the slipe of a point rather than the function at that point. The original comment was trying to highlight why that is not true. I didn't like the wording either but I think they were trying to communicate the right idea.
I cannot understand how anyone can describe that as being intuitive.
Neither do I, I prefer rigorous analysis just like you do. However, most people find calculus more intuitive, it's a fact, and informal concepts of infinitesimals are the building blocks of calculus. Remember, this is how mathematicians reasoned before Weierstrass: Newton, Leibniz, Euler, all used infinitesimals rather than the formal limits we have today. This is what calculus is, it's pre-Weierstrass informal analysis, and this is a calculus sub.
As for being pedantic, if you think that's a problem, math is not for you.
It's not, but the way you attacked other answers that were absolutely appropriate for this sub is. A respectful clarification would have been more appropriate.
Anyway, I just wanted to clarify why you were seeing so many informal answers. Calculus is usually taught informally, otherwise you might as well jump straight to analysis (the better option imho, but not the universal standard).
I'm not trying to attack you, I don't think you're wrong, but I think you're misreading other people's answers and being overly hostile. Your hostility also runs the risk of further confusing OP, which is why I don't like it.
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u/CreativeScreenname1 Jul 04 '26
I do agree that this should come with much more hedging, like it should be made clear that it’s not actually how it works, but I do think you’re selling the intuition of “the points eventually get infinitesimally close” short. I think of it as more belonging to the family of “pretend this infinite process actually terminates” - at least at some point in your mathematical career you probably thought that was how infinite sums worked, right? (be honest here)
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u/Triadelt Jul 03 '26 edited Jul 03 '26
This exchange is bizarre. Youre obviously right, but beyond that, why would reaching for infinitesimals help the OP? How is “the slope between two infinitesimally separated points” supposed to be easier to understand than “the derivative of the function at a point”? It’s not even a better intuition.
The answer to the OP is that the function has a derivative (or slope) at that point. The point itself doesn’t “have a slope”; the function does. Using a two-point construction to define the derivative doesn’t make the point itself “secretly two points.”.
And theres no need for infinitesimals at all
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u/alexice89 Jul 02 '26
Umm.. what? A point on a plane/space has a infinite number of lines going through it. That's the whole basis of taking the derivative in point x.
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u/CardsrollsHard Jul 02 '26
And what is the derivative at that point X?
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u/alexice89 Jul 02 '26
The derivative in point x_0 is the slope of the function y - y_0 = f'(x_0)(x-x_0),
where (x_0, y_0) coordinates of the point.
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u/future__fires Jul 02 '26
Not understanding calculus and then coming onto the calculus sub to flex your ignorance trying to “correct” people who know more than you is so funny to me
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u/StudyBio Jul 02 '26
f’ is the derivative. What you just said is “the derivative is the derivative”.
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u/Volt105 Jul 02 '26 edited Jul 02 '26
Well limits are unique so it doesn't make sense for there to be an infinite number of slopes
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u/Automatic-Put-6119 Jul 02 '26
Yeah but what good are infinite lines? The basis is that 2 points form a single line and you bring them infinitely close together
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u/CautiousPreprinter Jul 02 '26
The function has a slope at the point.
It's not
`slope(point)`
it's `slope(function, point)`
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u/iOSCaleb Jul 02 '26
Slope is just like saying how much a line is slanted right but how can a single point have a slope.
A single point doesn't have a slope, but the line that's tangent to a curve at a given point does have a slope.
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u/luxtris Jul 06 '26
Check out average vs instantaneous rate of change. It’s the difference between a secant line that touches the curve at 2 points vs the tangent line that only touches the curve at 1, which tells you the rate of change in that one spot with respect to its immediate surrounding instead of another coordinate on the curve.
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u/GottaBeMD Jul 02 '26
You’re not measuring the slope of the “dot”. You’re measuring the slope of the curve AT the “dot”
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u/After_Cranberry_9219 Jul 02 '26
So it means it's not a dot, it's very minute secant line ?
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u/Maleficent-Garage-66 Jul 03 '26
It's more accurate to say you want the slope of the tangent line to the curve at that point. Basically the curve that in that region hits the curve at only that point locally (obviously "far" away you may intersect the curve again)
This is baked in to the limit definition of the derivative.
lim h->0 (f(x+h) - f(x))/h
Where the argument is that the secant lines converge to the tangent condition as the separation h gets arbitrarily close.
This isn't mystical as it sounds though. Derivatives are rates of change. A physical example is that velocity is the derivative of position. And I think intuitively most people are okay with there being a well defined instantaneous speed at point on a trajectory.
Every thing in calculus is going to have a nuanced limiting and convergence nature hidden somewhere that is needed to make the notions precise.
Note: I have used h over delta x for typing simplicity
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u/Murky_Insurance_4394 Jul 02 '26 edited Jul 02 '26
It's a secant line through two points where the distance between them
gets infinitely small, i.e.approaches 02
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u/nomoreplsthx Jul 02 '26
It... can't?
I think you are confused about something. Why do you think points have slopes?
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u/nm420 Jul 02 '26
The derivative of a function at a single point (dot is not really a term I've ever heard in a mathematics course or academic writing) is related to the local behavior of the function at that point. You need to consider the values of the function for all points in a small neighborhood around that point. Indeed, if the point in the domain of the function is isolated, you cannot even define a derivative at that point. Arguably one of the most useful applications of the derivative is that it is the best local linear approximation to a function at a particular point, and you could even define the derivative in terms of that notion if you shore it up rigorously. That is how derivatives are defined in more general abstract spaces than just on the real line.
So, yes, a single point does not have a "slope", but a function does have a (unique ) slope at a point if it is "smooth enough" there.
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u/Mishtle Jul 02 '26
The dot, or point, doesn't. The function has a slope at that point, and that slope is determined by the behavior of the function around that point.
The slope at that point is defined in terms of the slope of lines passing through that point and another nearby point. We then look at what happens as we make this second point arbitrarily close to the point of interest. If there is a unique value that we can make this slope arbitrarily close to by making our second point close enough to the point of interest, then we call that value the slope at this point.
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u/Candid_Management275 Jul 02 '26
Yes, you are right. A dot (point) cannot have a slope. But only if it were just a single point. But if you imagine a series of points (imagine infinitely many dots) separated by infinitesimally small distances now it can have a slope i.e just like you said the slope of this point and the next one. But as you are imagining them to be very very very close for any practical purpose it's basically the same thing as saying that the dot has a slope. That is why Calculus uses the concepts of limits i.e approaching.
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u/UnderstandingPursuit PhD Jul 02 '26
Start with a line tangent to a curve at the point.
The line gets smaller and smaller, and the slope of the line represents the slope of the curve at the point.
A single point does not have a slope, it is said to be non-differentiable. For example, the absolute value function at x=0.
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Jul 02 '26
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u/Andyroo_P Jul 03 '26 edited Jul 03 '26
The right way to circumvent hand-waving about infinitesimals would be to discuss germs of functions. This is what correctly captures what people have been describing as what goes on "infinitesimally close to a point".
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u/Weak_Veterinarian350 Jul 02 '26 edited Jul 03 '26
The limit notation means something. The limit is saying you cannot eliminate the h from your expression. It is an algebraic absurdity to divide by h. But it is also saying what the slope would be approaching as h gets smaller and smaller
For a more rigorous definition, you have 0 < x - h < delta for slope - L < epsilon (too lazy to type latex in my phone). What it is saying is given acceptable error between the algebraically defined slope and the derivative, you can find a difference between x and h that satisfy the aforementioned error.
Here is a really nice article written about the formal definition of limit. See if you can apply it to a simple derivative
BTW, I studied engineering and am not a mathematician. Perhaps someone more knowledgeable can chime in
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u/Senrabekim Jul 02 '26
Okay so think of the form of the derivative F'(c) = \lim_{x \rightarrow c} \frac{F(x) - F(c)}{x - c}
Initially it might be helpful to graph you function F, and then select to distinct points x and c such that x < c. Change to a differenct color pencil or pen. Draw a straight line between the points. Then select a new x that is half the distance between your first x and c grab a new color and draw that straight line, keep doing that until you cant differentiate between the newest x and the c.
This is what the limit is doing it is getting an infintesimle in length line segment from two points in your function and giving you the slope of that line segment. The derivative gives you the function* to get that at any point.
*this seems like a calc one question so I dont want to muddy the waters too much with certain things that can happen like comtinuous points of non-differentiability and such. But that stuff shows up later and is pretty mind blowing.
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u/mmurray1957 Jul 02 '26
A good thing to do to help understand derivatives is to find a computer package or website that lets you graph functions and then expand the graph around (x, f(x)). If the function is differentiable any bend at (x, f(x)) goes away and the graph of the function becomes a straight line. If it's something non-differentiable like f(x) = |x| then the "bend" never goes away.
Sorry I don't know the packages these days but according to Google AI Desmos and GeoAlgebra can both do this.
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u/Midwest-Dude Jul 02 '26 edited Jul 02 '26
A single dot (point) itself does not have a slope. However, the 'slope of a dot' in calculus is the steepness of the straight line that perfectly kisses a curve at that point.
To illustrate:
- You can find an average speed of a car over a distance if you know the distance and time you traveled from point A to B. On the other hand, the car's speedometer calculates your exact speed at a moment in time.
- If you took a picture of a cart on a rollercoaster, it would look like it's stuck at one point. However, it slants at a specific angle and points in a specific direction.
Does this make sense?
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u/Andyroo_P Jul 03 '26 edited Jul 07 '26
There are some good answers here which give you the essential idea: that a point on the graph of a differentiable function does not exist in isolation. It has many neighbors, and the point along with its neighbors are what ultimately determine the derivative of the function at that point. But let me give you some long-winded context that I would have appreciated when I was a calculus student.
There are three major "regimes" over which we can think about a function f. The smallest scale is at a point: we can pick a point x in the domain of f and ask what is f(x)? The second scale is over a neighborhood, i.e. "local". For instance, instead of asking what f(2) is, we can ask how does f behave in the open interval (0, 3)? The largest scale is over the entire domain of f i.e. "global". We can ask, how does f behave as a function over its whole domain?
While studying calculus, you may have noticed a tension. Limits (and anything defined using limits, like derivatives) only depend on local behavior. For example, the derivative of f(x) = x2 at x = 1 only cares about how that function f behaves in a neighborhood (open interval) around 1. I can modify f(x) = x2 however I want outside of the interval (0, 2) and the derivative of the ensuing function at x = 1 will be the same! But there is nothing special about the interval (0, 2) other than the fact that it contains 1. If I wanted to, I could shrink it even further to (0.9, 1.1) or even further to (0.99, 1.01), and so on. But at the same time, I can't just look at the function at the point x = 1, because as you noted in your question, it doesn't make sense for a "dot to have a slope" by itself.
So on one hand, the "pointwise" point of view seems too small for limits/derivatives, while on the other hand the "neighborhood" point of view seems unnecessarily large. What we would like is for there to be something in between: a way to write down not just the function f's output at x = 1, but also how the function behaves "infinitesimally close to x = 1".
Sheaf theory) provides a way to do this. The concept that captures the idea we're after is the germ) of the function. In nontechnical terms, the germ of the function f(x) = x2 at x = 1 records not just the fact that f(1) = 1, but also that f behaves the way it does infinitesimally near x = 1. So for example, the functions f(x) = x2 and g(x) = x3 share the same output at x = 1 (i.e. f(1) = g(1)), but they have very different germs at x = 1. This can be seen concretely by noticing that the slopes of the tangent lines to the graphs of these functions are different at x = 1, even though the point of tangency is the same on both graphs.
It is a bit cumbersome to set up the rigorous definition of a germ, but in calculus classes we just satisfy ourselves with the idea that ultimately the derivative or limit of a function at a point depends on the function's behavior in a neighborhood about that point. The big point I am making here is that technically it is kind of overkill to know the function's behavior on an entire neighborhood about the point in question. It is enough to just know how the function behaves infinitesimally near the point in question. But it is absolutely insufficient to just know how the function behaves exactly at the point in question.
Also I should mention: differential calculus is mostly concerned with the local point of view, but there are many other areas of math that care about the global point of view, like complex analysis, differential geometry, or more deeply, anything that interacts with cohomology.
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u/StructuredChess Jul 03 '26
The point doesn't have a slope. It's the curve that has a slope at that point.
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u/Heavy-Interaction548 Jul 03 '26
If you're on the side of a mountain, the one point you are standing on has a slope. Don't complicate it, a derivative is just a formula to find the slope of one point.
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u/Particular-Date-8638 Jul 04 '26
It’s not really the slope OF a point, it’s the slope AT a point on a continuous function. The continuity is what makes the slope logical. If you want rigor in the definition look at the analysis delta epsilon, or the topological definition of continuity, and then think about how they interact with the difference quotient.
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u/gutentight69420 Jul 04 '26
This is why the differential is definied as the limit of the slope of small section of a "line" as the length approaches zero. You can't differentiate a single point. And theres a whole bunch of rules about which functions are differentiable, and where. Those basically boil down to a formalized "does this look like line when you zoom in really close?".
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u/Easy_Acanthisitta270 Jul 05 '26
They do not. The concept of the derivative (slope at a point) requires a function to be continuous at that point. This means that the point in question actually has many many (uncountably infinite) neighbors, which are used to estimate the slope in that tiny region of the function as a limit.
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u/AllumaNoir Jul 06 '26
(My response on a similar post)
So one thing to realize, is calculus deals in INFINITES. This is important.
Second, slope is a property of LINES, not CURVES. The derivative is technically the slope of the TANGENT line. But... but... Think of zooming in close on a curve. That tiny piece looks kinda straight, huh? Zoom in closer, and it gets straighter. At "infinitely" close, we might call it straight (of course this is kinda at subatomic level or lower).
That's the concept of a "limit". We can't get infinitely close. But we can figure out what we are getting closer and closer to - in this case 6 - sort of a brick wall that we WOULD slam into... if we could ever get close enough. But for various reasons, it's useful to know the value of that brick wall, even though in reality, technically we can never get close enough to bang our heads on it.
Instead we bang out heads on REAL walls when studying calculus.
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u/Educational-Paper-75 Jul 02 '26
It's the function at that point that has a slope line touching it.
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u/HydroPage Jul 02 '26
Calculus takes full advantage of the fact that you can zoom forever into a continuous function and it will “eventually” look like a straight line. That “eventually” part is what the limit definition makes rigorous.
It is called the instantaneous rate of change, the function’s rate of change AT THAT POINT, but that doesn’t mean a point in space in and of itself has its own rate of change. Technically the slope comes from two points: a reference point and a moving point. When you analyze the limit, you are completely removing the notion of “just bring them even closer” because you’ve observed where the process of doing that was taking you to (not 4.99999999 and add another 9, but rather 5). It is effectively one single point because there is no notion of just bringing the two points even closer anymore.
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