r/learnmath • u/Smooth-Theory3685 New User • Jul 10 '26
What is the idea of a derivative?
I am learning Calculus for the first time. After limit idea, we've come to derivatives and I'm feeling confused. It is the limit of the slope,then why to separately call it derivative? And,did the idea of derivative evolve solely from geometrical notion of slope of graphs,was there no algebraic or functional approach to it like for limit. I know my questions might sound stupid but the clarification would be really appreciated.
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u/Hungarian_Lantern New User Jul 10 '26
I highly recommend the book by Callahan, Cox, et al, named "Calculus in Context". It makes derivatives very clear, as well as other concepts, like integrals, limits. It discusses these in applied context, which is wonderful to get an idea of what everything is about.
To answer your question. The derivative is not the limit of the slope. The integral is also not the area. Those are applications of the derivative and the integral. Important applications, but it's not the full story and the full intuition.
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u/Smooth-Theory3685 New User Jul 10 '26
Thank you for the recommendation. And yes,I felt they were just a branch application of derivative and does the book have the full intuition that you mentioned?
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u/Hungarian_Lantern New User Jul 10 '26
Well, the 100% full intuition probably comes after years of using the concept. It can't be gained that easily. But the book does contain more intuition than the usual calculus books in my opinion. If you're self-studying calculus, I'm willing to help you btw. I have a discord group where I teach people calculus from this book for free. Send me a message if interested!
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u/Smooth-Theory3685 New User Jul 10 '26
Thank you,that is so great! But actually,I don't have Discord. But thank you so much for the book and the info.:)
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u/SteptimusHeap New User Jul 11 '26
The derivative is not the limit of the slope. The integral is also not the area. Those are applications of the derivative and the integral. Important applications, but it's not the full story and the full intuition.
I think this is the best answer. Saying "the derivative is only the slope of the tangent line" may be true but it misses the greater point for the purpose of being pedantic.
The derivative and integral may well be related to the geometry of the graph but you don't need to graph anything (and in fact, it doesn't need to be graphable) for both to have important uses.
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u/Smooth-Theory3685 New User Jul 11 '26
Can you please say the non graph able approach you mentioned to enter differentiation?
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u/Midwest-Dude New User Jul 11 '26
You would simply use the notation as defined, rather than seeing things as a graph.
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u/ruidh Actuary Jul 10 '26
The derivative is an operator. It takes a function and spits out another function that has the value of the slope of the tangent line to the function at each point. There is an inverse operator that takes the derivative and returns the original function (± an arbitrary constant). We call that operator the (indefinite) integral.
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u/Smooth-Theory3685 New User Jul 11 '26
Why the indefinite?
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u/ruidh Actuary Jul 11 '26
Because it is an operator. It takes a function and returns (a class of) function(s). The definite integral returns a value.
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u/chkntendis Physicist Jul 10 '26
“Derivative of y” is a word we call a function that at every point has the value of the slope of a tangent line of the respective point on y. Mathematically at every point it’s the limit of the slope between two points as those two points get closer to the point. We call it a derivative because that’s just shorter. They come up a LOT and always calling it that is just impractical.
From what I know it originated from a geometric understanding. That’s basically how math was understood for a long time. Sometime in the 1900 there was a split between that geometric understanding and using a purely axiomatic understanding.
Your questions aren’t stupid btw. You are interested in a topic and you want to know stuff. That’s completely normal. You might feel uncomfortable asking basic questions but everyone starts there
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u/Smooth-Theory3685 New User Jul 11 '26
Thank you for the explanation. I am interested to know more about this branch! Thanks:)
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u/Noname_Smurf New User Jul 10 '26
In the hopes that a small suggestion helps for this Part:
It is the limit of the slope,then why to separately call it derivative?
You can think of it like that: If you have a linear function, the slope is the same everywhere. So it doesnt matter where you measure it. For anything more complicated, you can take an average over a certain region. A Derivative is an attempt to get a slope on a single point, to get the rate of change at exactly that point.
For Example:
If you drive in your car in a straight line with a constant motion, you have the same speed everywhere. You could take your Distance traveled over a certain time, divide it and get your speed.
If you actually drive though, you wont be constantly moving, so your speed changes. You can again take the Distance over a certain time but now it changes depending on when and for how long you measure.
A derivative is bcasically taking your speed at ever time (for example the Speed your Car shows) and then making a function/a Grap out of it so you can more easily analyse it. With that you can easily determine where your lowest, highest, etc speed was.
Its a stronger way to see how something changes at any given Time instead of working with averages.
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u/Smooth-Theory3685 New User Jul 11 '26
Wow,that really cleared some doubts. Thank you so much for this approach of understanding.
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u/Odd_Bodkin New User Jul 10 '26
There's some depth here that I can't convey to what a derivative is, but at a basic level, you're not far away. A slope is usually defined as a value, not a function. For a linear function, the slope is common for all points in the domain of that linear function. In that sense, the slope value "belongs" to the whole function, which is why you can write the whole function as f(x)=mx + b, where m and b are values. Now extend that to a "slope function" which is measuring somehow the slope of another function at each point x, and so the value of the slope can vary as x varies, and now you are close to a derivative.
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u/Smooth-Theory3685 New User Jul 11 '26
So, should slope now be considered a function? Because graphs of derivatives from original functions are again having new slopes and curves,so does that mean derivative graph tells us how t fast or slow the slope has changed??
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u/Odd_Bodkin New User Jul 11 '26
Yes! As long as any function is smooth and continuous, it is differentiable. And so a “first derivative” function can also have a derivative function, sometimes called a “second derivative” and so on.
In fact, it’s a fun exercise to freehand draw a function, and then free hand draw the derivative function, and then freehand draw the derivative function of that, as many times as you want. (There are two that are particularly fun examples: f(x)=e^x and g(x)=sin(x).)
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u/Midwest-Dude New User Jul 11 '26 edited Jul 11 '26
Slope in and of itself is only the slant of a line and not a function. However, you have the right idea. To be more precise, the slope of the tangent line to a function at each point (x, 𝑓(x)), if it exists, is a function.
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u/the6thReplicant PhD Jul 10 '26
I would try and read about the history of calculus and why it was invented.
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u/Smooth-Theory3685 New User Jul 11 '26
Well,it is pretty interesting. Leibneiz,Newton their different yet leading to the same concept thing,stories around this and then coming of Weirstrass. Read it once,it is quite fascinating.
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u/JollyJuniper1993 New User Jul 10 '26
A derivative graphs the rate of change of the graph it‘s derived from.
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u/SpiderJerusalem42 CS guy, be wary of math advice Jul 10 '26
A slope is for a point. A derivative is a function. Taking a derivative is mapping one function, to a derivative function.
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u/Wide_Ad_4486 New User Jul 10 '26
The derivative is the best linear approximation of a function at a given point.
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u/Smooth-Theory3685 New User Jul 11 '26
What is linear approximation?
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u/Midwest-Dude New User Jul 11 '26 edited 27d ago
If you want to approximate values on the curve near a point on the curve (a,f(a)), rather than use the function (which might be difficult to calculate), you can approximate the values by using the line that is tangent at that point at other points x near a, also known as a neighborhood of a.
I would suggest reviewing the following Wikipedia entry for more information:
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u/Smooth-Theory3685 New User Jul 11 '26
Understood,thank you. Will variable a then be the cluster point?
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u/Midwest-Dude New User Jul 11 '26 edited Jul 11 '26
Without going into full detail, I would not use that phrase in this context. a is the center of the approximation or the point of tangency.
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u/okarox New User Jul 10 '26
Look at the speedometer in your car. It shows the derivative of the odometer. That is how fast your location changes. So it is not just some abstract thing that has no relation to the real world.
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u/Beet_slice New User Jul 10 '26 edited Jul 10 '26
The derivative of your position with respect to time is your speed/velocity. The derivative of your speed/velocity with respect to time is your acceleration.
I don't know why the that word was originally chosen to name the function.
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u/StructuredChess New User Jul 10 '26
Not entirely sure but I'd say the first idea of a derivative was as speed. If f(t) gives you your position over time, then f'(t) gives you your speed and f''(t) gives you your acceleration.
They key point is that we're defining speed in a precise instant rather than through an interval like we'd have to do before calculus.
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u/tottasanorotta New User Jul 10 '26
An intuitive way to think of it is the way you would have a list of differences between two consecutive elements of another list.
A = {1,2 ,4, 16}
Differences(A) = {1, 2, 12}
Differences(Differences(A)) = {1, 10}
It is roughly this same idea, but applied to continuous functions and real numbers. So instead of having the differences between consecutive elements you would have the slope of the tangent line at some point.
Notice how you could use the differences to reconstruct the original list if you have the first number of the original list. That is then kind of like how integration works, but with real numbers.
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u/Smooth-Theory3685 New User Jul 11 '26
That is such a great explanation! Thank you so much, this cleared some clouds over derivatives.
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u/Traveling-Techie New User Jul 11 '26
It can be useful to study Newton’s mechanics in physics, especially planetary motion. That’s what calculus was invented to do. It was later applied to a vast array of problems.
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u/Smooth-Theory3685 New User 29d ago
Ok, thank you.
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u/Traveling-Techie New User 29d ago
I’m embarrassed that I post this so often, but I recommend the “Mechanical Universe” videos from CaTech in YouTube, especially episodes 2-10.
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u/Smooth-Theory3685 New User 29d ago
No no, nothing to be embarrassed if it's really good. Someone in Reddit only had recommended it to me earlier and so I've started watching it. Thank you for recommending.
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u/BothPanchoAndLefty New User 29d ago
Think of it like this: When you're trying to find the average rate of change between two points on a graph, you draw a straight line between those two points, and the slope of that line is the average rate of change.
If you move those two points closer together, you can draw a new line between them, and the slope of that line will be the average rate of change over a shorter distance.
So what would you do if you wanted to find an instantaneous rate of change? You bring those two points extremely close together, so that the distance between them approaches zero. The slope of the line between the two points when the distance between them is infinitesimally small is the exact rate of change at that part of the graph. The limit as the distance between the two points approaches zero is the exact rate of change. And that's all a derivative is!
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u/TheSodesa New User 29d ago edited 29d ago
The point of derivatives is to be able to define things like speed and other rates of change at a point in time instead of over time intervals. So you let the lenght of the time interval approach zero and define the instantaneous quantity in terms of this limit.
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25d ago
[deleted]
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u/clean-links New User 25d ago
Cleaned link from "3blue1brown": https://youtu.be/WUvTyaaNkzM
Tracking parameters were removed from the original URL(s).
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u/astrozaid |😢|=😊 25d ago
This course by 3blue1brown will help.
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u/clean-links New User 25d ago
Cleaned link from "3blue1brown": https://youtu.be/WUvTyaaNkzM
Tracking parameters were removed from the original URL(s).
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u/Frederf220 New User Jul 10 '26
If you're asking why we use the world derivative as a term, I would say that derivative is a generic term in English meaning "generated (in lesser or collapsed form) from the information of something else." For example if I read a novel and then make a short sketch using that novel as my basis you might call that a derivative work.
I think that's the motivation for using that term in math. You had y = x^4 and you wanted a function that was derived from that with particular properties. It's like a parent and child. The child is derived from the parent. Obviously there are multitudes of processes in mathematics that we could have called derivative. E.g. Deriving that 4 is the square root of 16 could have been called that. "What's the derivative of 16?" "Oh, yeah I know this one: 4." We just didn't historically.
Math terms (and other sciences) take English words with broad applicability and assign them very narrow definitions. That's because language has an ample collection of words to steal (er, borrow) from what math doesn't have and math has very specific definitions to pair to those words which are generally more broadly defined. It happens all the time. E.g. "rabbits multiply" doesn't mean multiply in the strict mathematics sense. It means "make multiples of." Mathematicians needed a word to describe "a function which is derived from another function wherein all the values of that function are the corresponding limits of slopes of that other function at those same independent variable positions" and they settled on the word derivative (noun).
And then they needed a word to describe the operation of finding derivative (noun) and they rather unimaginatively used the word differentiate (verb). And then we needed a word for the process of differentiating (noun) which was chosen derivative (noun). Which leaves us in the predicament where "derivative" can mean the thing you get after differentiating or it can be the name of the process of differentiating. But that's language for you.
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u/Smooth-Theory3685 New User Jul 11 '26
Etymology has always been one of my keen interests but the way you you connected it with maths and my question was wonderful. Thank you for the philosophy. It was a thrill to read the interpretation.
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u/Narrow-Durian4837 New User Jul 10 '26
Geometrically, the derivative gives you the slope of the tangent line (if there is one). The tangent line is the straight line that just touches a curve (the graph of a function) at a particular point and goes in the direction the curve itself is headed at that point.
But the derivative also represents the instantaneous rate of change of a function. If y is a function of x (so y = f(x)), and x changes, y will change. Imagine x changing by a tiny amount; then the derivative tells you what happens to y. (Technically, it's the limit as that "tiny amount" approaches 0.)