Calculus: Building Intuition for the Derivative

How do you wish the derivative was explained to you? Here's my take.

Psst! The derivative is the heart of calculus, buried inside this definition:

This is a companion discussion topic for the original entry at

@Sebastian: Thanks, glad it helped :).

Aha moment came near reading your fab articles must write a high school book soon I wanna read more and more. ,. Plzzz do write on physics too.

Hi Ansh, glad you enjoyed it!

Aha moment came near reading your fab articles must write a high school book soon I wanna read more and more. ,. Plzzz do write on physics too.

I find many pluses and minuses with these types of approaches to topics. It’s good because some people find it more approachable. I feel it’s bad if they are not able to convert it to a logical mathematical understanding using mathematical language.

This is one of the most harmful aspects of math education today. Everyone is focused on pushing standardized testing and standardized testing destroys the nurturing of problem solving and logical understanding of concepts because there is no time and everyone has to play the ‘rat race’ within education to get the ‘golden ticket’ to a nice expensive college. This is what you get when you turn education into a ‘product’ and students into ‘consumers’. Those students who have excellent problem solving skills never have time to nurture them and even wind up having those skills stunted.

My rant aside, no, the slope of the tangent line is not a bland description at all. The problem here usually is that students don’t have a solid foundation in algebra first, which is a must and then a very good foundation in pre-calculus.

To see a whole topic on derivatives without a single graph is doing a disservice.

Good to see students get something here but calculus needs a unified approach and the understanding of the derivative begins with a strong foundation in algebra (coordinate geometry) and pre-calculus.

I’m uncertain about calling the derivative a “better division”, although it’s better than “continuous division”. I’d probably call it “generalised division”. It does follow the pattern (established with integrals) that the derivative is about changing quantities. I believe the problematic aspect of the derivative, is that it is a number at a specific point ‘a’, f’(a), but a function at a generalised point, f’(x).

I do like the 4-step procedure: (1) Choose an interval; (2) Find the raw change; (3) find the rate of change; and (4) Make your model perfect. But the limit is not only about making your model ‘perfect’, because it is also used to simplify a problem by neglecting the contribution of a certain component.

That last step “Make your model perfect” seems to be what the change from Hyperreal numbers to Real numbers (by taking the standard part) is all about. Or at least that was something that immediately sprung to mind.

Argh. Somebody mentioned the epsilon – delta definition. It’s not so much the definition itself (that basically relates input error to output error), but most explanation are just so…ugh.


kishore, the differential equation doesn’t come from a function. It comes from a model that predicts the observed values. The model happens to imply certain relationships between physical measurements, which (when stated mathematically, using known laws of physics, and expressed with the smallest number of independent variables) often winds up having integrals and differentials embedded in it – hence it’s a differential equation.

i need more details on how to solve the partial fractions and integrations.

It is realy interesting. I have enjoy it …nd lear a lot. Today I unmderstood What is Derivative ? Actually I am searching this but give us. Than you so much. Please give the this opportunity to learn math.

I dont understand cumulative frenquency so well.pls help

What a great explanation! It took me back to my days as a Physics major in college - only I wish I had this explanation back then :slight_smile:

No problem, I’ll be here.

And if it helps any, the playlist I pointed you to is “MathFundamentals”, so it’s no problem not knowing math. He starts at counting with tallies, if you want to start at the beginning.

If you wanted an advanced playlist, he’ got one on universal hyperbolic geometry and another on algebraic topology. Whew! :slight_smile:


Ok Tanksley, I’ll check out the videos, and then I’ll post what I’ve understood. But since I am unfamiliar with a lot of what is being discussed here, I’ll need your confirmation to be sure what I understood is correct. I’ll wait for your comment.

And thank you for pointing me to these videos. It’s a new approach for me - integrating algebra, geometry and calculus. The only hindrance is my own less rigorous math background. So I’ll have to go through it step by step. I hope you will stay on the site to comment on my progress.

[…] The derivative f’ (df/dx) is the moment-by-moment behavior […]

one thing that always bothers me is the chicken egg problem. which comes first, differential equation or the function. Let me give and example. Take for instance decay laws which is stated in differential equations. But when you carry out practical experiments, we would plot a graph and would approximate the graph to a function through curve fitting techniques. Now where is differential equation fitting in.Because i can make all the predictions through a function. What is the point in representing event through differential equations if my function could do all the job.

I’m really sorry, but I’m just not able to get the time to reply this weekend. You’re on the right track in general (in fact, I’m quite impressed, given the tiny bit of explanation I’ve been able to give); but there’s more to do.

If you don’t mind, I’m going to point you to a YouTube video where a fairly complex curve is analyzed according to these rules.

Unfortunately, he uses some unusual terms while doing this – for example, he denotes the curve using a “polynumber”, which he writes as an array of integers. You may be able to figure how a polynumber is like a polynomial without explicitly written variables; if you need a better explanation the previous videos in his series will explain completely. See the entire playlist at:


@AK: Thanks for the comment – really appreciate the support! I’m actually looking at ways to help tap into the community – one idea is getting a little section after each post to share the analogies that worked (or questions that are still outstanding). I’d love certain articles (like the one on e, for example) to become a living reference about “What actually made it click”. Wikipedia is great for strict definitions, Khan and others for detailed tutorials / practice problems, and I’d like to contribute aha! moments (i.e. the last step that turned the light bulb on). Definitely something I’m looking to develop, I’ll be posting on this soon =).

There’s a mistake in my above post, I said “Statement 1 should be – the tangent is the best first order approximation to the curve at a point ie it should have the same value as the original curve and also the sam rate of change at that point.”

This is what one would expect given the traditional definition of the tangent.- ie the tangent line to a plane curve at a given point is the straight line that just touches the curve at that point.

But if you look at the equation to the tangent line derived in my last post, y1 = 3a^2x + a^3,
at x = a, y = 4 a ^ 3. The point (a, 4 a ^ 3), does not lie on the curve y = x^3.

So is there something wrong with the definition, or is there something I’ve missed ?

Tanksley, I went through your last post again and I think I am beginning to understand the definition of the derivative as the slope of the tangent.

Statement 1 should be - the tangent is the best first order approximation to the curve at a point ie it should have the same value as the original curve and also the sam rate of change at that point.

So, if my function is x^3, and I want to draw a tangent at

change in y = (x + a)^3 - x^3

              =3a^2 x + 3a x^2 + a^3

The zeroeth order approximation is found by setting the first and second degree terms to zero. This would be y0 = a^3.This has the same value as the function at x=a.

The first order approximation is found by setting the second degree terms to zero. This would be y1 = 3 a^2 x + a^3. The slope of this line has the same value as the derivative of the function at x = a ie 3 a^2.

Intuitively too, this makes sense.Let’s consider a body starting from rest (at t = 0) and undergoing uniform acceleration of 1m per second squared. When I say this body has an instantaneous velocity of 8m per second at t = 8, what it means is that the body has a potential to travel 8m per second, if it were moving at a constant velocity of 8m per second, in either direction. (But this doesn’t happen, because by the time the t becomes 9, the body has already accelerated through 1m per second squared. So the distance travelled between t = 8 and t = 9 is not 8m.)

Is my understanding correct, or is there something that I’ve missed ?