Introduction to Calculus with Derivatives | Hacker News Reader

Introduction to Calculus with Derivatives

adit.io

266 pointsnafizh65 comments

If one wants to learn calculus, I can't recommend Silvanus P. Thompson's Calculus Made Easy enough.

It's one of those old (1920's, IIRC) books that gets right down to brass tax, and doesn't clutter your understanding with useless fluff designed to make learning maths "fun". These kinds of books are all too rare.

Found an html conversion - http://calculusmadeeasy.org/

This was one of the books Feynman used to learn Calculus.

The other was Advanced Calculus by Woods:

https://libgen.pw/download/book/5a1f04723a044650f5fbaf91

I have seen this discussed online several times. Here is a HN thread:

https://news.ycombinator.com/item?id=14192894

that indicates the text Feynman first used was Calculus for the Practical Man by J. E. Thompson. Both are quality texts and as the linked HN thread notes, can be found on archive.org.

My mistake. Thanks!

1914 to be picky: https://news.ycombinator.com/item?id=14161876 (link to pdf)

It's also on Project Gutenberg: https://www.gutenberg.org/files/33283/33283-pdf.pdf

Looked at the web version. Wow, it was nicely and clearly written. Clear, simple, and most importantly to the point.

Can someone please recommend more books like this?

(The subject matter does not matter much, it is a joy to read such books regardless of actual subject!)

Hah! I'm reading this as well, and it's not only hilarious to read, but man, it's so easy.

This explanation actually reminded me of that book and I suspect it was an inspiration.

I couldn't find contact details, so I have to reply here.

For reference, the expression is "Brass Tacks":

https://en.wiktionary.org/wiki/get_down_to_brass_tacks

When I was on vacation a couple weeks ago, I was inspired by the Feynman lectures on physics to review basic calculus.

This time I tried my best to understand derivatives at a low/basic level. When I was in college, many years ago, I was doing my best to just to tread water in physics and the early stages of the calculus classes.

I did my level best to try to explain derivatives as if I knew nothing of calculus. Used my wife as a guinny pig. What I ended up with explained both derivatives and integrals at the same time in a very basic, but correct (I believe) sort of “add up the arrows” sort of way.

I really struggled to do this, but I thought it was really cool when I finished. Well finished is perhaps too strong a term. Most of it is scrawled out in handwriting on my iPad.

I’ve been meaning to go back and wrap it up and put it on the web. I think I’ll try tonight and post back here or somewhere.

Just read through the post to make sure there was still value to me doing this. Seems like it. We start similarly, which is good. The visual description seems to be pretty different so I think there will be value in another point of view.

Don't hesitate to do that!

> guinny pig

Guinea pig. They were originally believed to have originated from west Africa (Guinea), hence the name.

I did this today with neural nets and training classifiers. Looking forward to reading your notes!

well i really hope you do, i'd love to flip through it. each time i read some other person's notes, another tiny angle of light hits the statue.

"That change c never truly goes away, but it is so small that it doesn't really matter: if you make c small enough, the derivative is basically 8. So we're just going to pretend that all the cs are so small they have become zero."

My first calculus teacher taught us derivatives in a similar way - but I have to say that, for me, this imprecise language confused me. When and why is it OK to pretend that c is zero!?

Further on, the official definition using a limit claims that the "c->0" means "make c as small as possible". But that's not what it means. Again, this is imprecise language that confuses more than it explains. What does "as small as possible" even mean? If we want to make it as small as possible, why not set it to 0!? How small is small enough?

I think somewhere in this article there should be a precise definition of what a limit is, using epsilon and delta.

It should be saying something using the words "arbitrarily close" and "sufficiently small".

Something like : "we can make the approximation on the left arbitrarily close to the expression on the right by choosing any value of c sufficiently close to 0, even though the expression might be undefined when c=0. Epsilon: you tell me how close you want the approximation to be. Delta: I tell you how small c needs to be".

There used to be a great website that was precise but also informal: karlscalculus.org - but sadly it appears to be down now.

I think it was Weierstrass who first gave a proper definition in terms of epsilon / delta (e,d). To say it more precisely:

We say that the limit

lim x -> c f(x) = L,

if for all e > 0, there exists a d, such that for all x, if

| x - c | < d,

then

| f(x) - L | < e.

The history seems to be summarized well in this wikipedia entry: https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of...

The rigorous definition of infinitesimals is developed in non-standard analysis: https://en.wikipedia.org/wiki/Non-standard_analysis

IIRC you define e such that e^2 = 0 and then normal algebra just works.

There is background work needed to precisely concepts but that's the gist of it (again, IIRC).

> Further on, the official definition using a limit claims that the "c->0" means "make c as small as possible".

Yeah.. I’m pretty sure the official definition of a limit says nothing like that.

Any definition of a limit that I’ve ever seen (official or not) has been careful to note that you cannot set c to 0, but only approach it (whether in mathematical terms or English).

Yes, that's my point. The article is giving an incorrect definition of what a limit is. The target audience does not know what a limit is or how it is defined. The use of imprecise and incorrect language will confuse these people more than it will help.

Found a link on the Web archive: https://web.archive.org/web/20080117115912/http://www.karlsc...

Essentially from the definition of the derivative you can write

f(x+h) = f(x) + h f'(x) + O(h^2).

If you use this to solve differential equations its called the Euler method.

> its called the Euler method

Disclaimer: and it sucks

I like this take on introducing derivatives.

My first intro to derivatives was a little less than 20 years ago, but I feel like it was very much in the "traditional" vein of: Suppose we have "f'(x) = lim(h->0) (f(x+h) - f(x))/h" and we substitute in various equations. What will f'(x) be?

The difference as presented here: I (re?)learned an estimation method for decimal place mathematics while at the same point tying it to a larger/underlying principle.

I think a great approach would be to then do the stuff I started with, e.g. finding f'(x) given f(x).

Out of curiosity, how many of you have seen the approach as seen in the above article? I can't recall seeing it before, but again, it was a fair time ago for me.

Just have to add K.A. Stroud's Engineering Mathematics to this list. Easiest maths book I've ever known, with only the slightest amount of hand waving. If you're having any trouble with it, this is your go-to book.

Unrelated to calculus, a simple mental trick for quickly calculating squares close to round numbers is to just break it into (a+b)^2=a^2+2ab+b^2. E.g. 51^2 = 2500+100+1=2601.

This is a slight reduction of the FOIL method, a classic in any first year algebra course. A decent trick to remember on tests when you are trying to quickly solve a lot of algebra problems.

Great article. Though I know some basic calculas but I never really thought about calculas this way.

This is an interesting and eye opening perspective. Wish to see more of such articles.

It looks like a very nice careful description, but where is the physical intuition? Where's the plots of tangent lines on a function at various points?

There's nothing wrong with the article (I think), I am just curious about the pedagogy and motivation for not immediately introducing a graphical conceptualization of a derivative especially for such a "from the ground up" exposition.

Hi, author here. This is what made derivatives finally click for me (as opposed to the tangent line) so that’s how I wanted to explain it...maybe the same thing would click for others!

I was looking for a tutorial on derivatives the other day and most of the results popping up at the top of Google were a bit dry.

This one is very good though. Excellent work.

How did you do the diagrams?

The Procreate app on the ipad

Do you have any recommendations for learning the app to make what you’ve made?

I don’t suppose I could bribe you into recording a video of yourself making some diagrams...

Your algorithms book is excellent and fun!

IMO it is more about drawing ability vs app knowledge, so I would suggest learning to draw! Just start drawing and you'll get better over time. You could join something like https://streak.club/s/8/daily-art to keep motivated.

The daily art streak was an awesome suggestion, thank you.

I have one specific question. When you write in your diagrams, what brush settings do you use? Are you using calligraphy mode?

The handwriting in your diagrams has a unique look and feel that I was hoping to emulate.

For example: https://i.imgur.com/8Bb1wEY.png

That question mark is beautiful, and the thickness varies in a particular way that is quite unlike regular handwriting. It's more like chalk on a chalkboard. That's why I was hoping to know the exact settings.

I spent some time with the Procreate app today and was able to pick up the basics. I'll try learning by copying your work. Thanks again!

Oh I see ... that's done non-digitally using a fountain pen.

For what it's worth, I like this much better. I had the traditional method of graphs and tangents. I was slightly lost on it in Cal I, and I could regurgitate it in Cal II. "The derivative of a function is the amount of change of the output with respect to the input." This is another level of derivatives clicking

And thank you for it. Tangent lines have never meant much to me; your way of explaining feels much more intuitive.

Also, I love your algorithms book! It is the main resource I used when preparing for my coding interviews.

Thanks again, I look forward to your next project!

Thank you!

Hi author. There is a typo in the section How To Calculate The Derivative. The first two images in that section label 0.1^2 as "input change" in the denominator. It should be 0.1, not squared.

Echoing the sibling comment: How'd you make the diagrams? They're quite lovely and I'd like to follow your process.

I do not really learn a concept until I can visualize it somehow. Indeed, when I was learning derivatives, it did not click until I saw a graph with a tangent line. But everybody is not the same way. Some people prefer to develop their intuition based on a text explanation, some people need an example with numbers, some other people need an actual person to explain it to them in spoken words.

When you learn better with one of these methods, seeing another one first may confuse you instead of helping. Moreover, once you have understand the concept, other kinds of explanations tend to broaden your view of the topic.

I think this a great non-visual explanation of derivatives. No more, but no less. I think it has a great pedagogical value, even if it is not the best introduction for my way of learning.

Way back when I was in high school what made derivatives click for me was the notion of "Rate of Change" applied to time series.

I think it is called Parametric Calculus.

v = u +at; v^2 = u^2 +2as;

and all that

Velocity = ds/dt and acceleration = dv/dt = d^2S/dt^2

At least in my mind rate of changes makes a lot of sense to explain derivatives.

edit: i.e Acceleration is the rate of change of velocity over time.

Reducing explanation of the derivative at an intuitive level to 'simply the rate of change' confuses the hell out of people when they encounter other things that are also defined as derivatives but do not describe a change in any obvious sense. For example, electric current (or, say, a flow of water) through a cross-section of a closed circuit does not necessarily represent a change of electic charge (or the mass of water) on either side of the surface, as it remains constant.

Isn't there a better, more clearer, and more intuitive notation for calculus than dx/dy, or even f'? I feel these notations are somewhat ambiguous and bordering on abuse of notation.

Yes. There is an operator notation for the derivative where you can both avoid the not-quite-division and specify the variable you're dealing with. https://en.wikipedia.org/wiki/Notation_for_differentiation#E...

I'm from Belgium, and here kids learn this in high school (5th year). It's basic knowledge. So why does this show up on HN?

Only kids who got into higher education tracks get calculus. No need to be smug about it, many kids don't get in those tracks for any reason and might find later in life that they are interested in calculus.

I feel if anything the US education system exposes to more children than the European systems, even though I feel the system is misguided.

In any case, I had calculus in highschool, then more of it in University and I still feel I could do with a fun refresher every now and then.

There are many paths that people take towards mathematical understanding, many paths have loops where material is covered over and over again at different levels of sophistication.

I feel that typical pre-college curriculums cover topics superficially-- just enough for the exams. Many students would benefit so much more if they held-off on calculus until they became fluent in "the basics". Instead you get students practically forced to take calculus in high school whether they're ready or not and then they need to repeat the material in college-- sometimes STILL under-prepared.

For college-bound, it would be much better to slow down, focus on rigor and mastery of algebra, geometry, practical applications, and proofs. Then in college start with something much deeper and more comprehensive than your typical "Calc 101"-- maybe at the level of Spivak's Calculus text or Rudin's.

> I feel if anything the US education system exposes to more children than the European systems

What do you mean?

Well, the way I understand the US system is that all children basically get the same education, and they are divided only by grades. In the NL and I think other European countries also, children go in separate highschool tracks depending on their performance in elementary school. A child that exhibits weakness in logic/maths/puzzle solving in the last year of elementary will likely be recommended to go into a vocational track and only ever receive basic maths education unless they opt in to a higher education track afterwards.

In the NL often these tracks are even in separate schools, but always separate classes. So my sister went to a different highschool than I did because I was showing more proclivity towards scientific education. After her highschool if she wanted she still could've opted for a track that would qualify her for scientific education, but since she had a preference for arts she went to an arts academy instead.

FWIW, your understanding is incorrect. While true that in US schools, all students are expected to complete a full 13 years of schooling (Kindergarten + 12 grades), students receive individualized courses of study. By high school, there may be 3-4 levels of instruction for Math ranging from remedial to college-bound, to college-level coursework. Additionally, non-college-bound students at most schools have the option of taking classes that prepare them for more traditional blue collar work (like automotive repair, etc) and often students have the option to go to a different school more aligned with their talents/skill level.

The primary difference between US and European education is in the US taking particular courses of study is the decision of the student/parents and not of the state.

Ah cool, thanks!

Yes, it's similar in Belgium. But, derivatives are really considered basic knowledge, so I doubt there's a track you can pick in Belgium that does not expose you to this. I will check.

update: combining www.onderwijsdoelen.be and the distribution of students across the different tracks, 69% of the students are required to be exposed to this.

I, for one, enjoy reading alternative explanations of difficult subjects. Calculus is not basic knowledge in my book. Ironically, my dad teaches high school calculus, in the US for whatever it's worth.

I'm from Italy and it's the same here, but the problem remains: it seems that a big part of population (devs are not excluded) have problems and hate relationship with math.

Yep. In Finland, derivatives are discussed (in a superficial manner) in high school even if you take as little math as you can, but from what I’ve heard most people who don’t elect to take more in-depth courses don’t really grok what they’re about and promptly forget about them after the exam.

I graduated college from the USA but with a liberal arts degree and the highest math I learned was precalc.

I think this is backwards. Explaining derivatives by Physics (speed or slope) seems a lot more intuitive to me.

Backwards for you - different people approach things from different points of view.

I was hoping this article was going to introduce calculus using financial derivatives in practical examples.

Instead the article asks how would one extimate 4.1 squared without a calculator.

I can't recall of any time in my life when I needed to do so. Now that I have a smart phone and watch, I don't anticipate needing to do so any time soon.

If everybody was of the same opinion as you, we would not have smart phones and watches.

Reply on news.ycombinator.com