
Introduction to Calculus with Derivatives - nafizh
http://adit.io/posts/2018-02-18-Introduction-To-Calculus-With-Derivatives.html
======
presscast
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.

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

The other was _Advanced Calculus_ by Woods:

[https://libgen.pw/download/book/5a1f04723a044650f5fbaf91](https://libgen.pw/download/book/5a1f04723a044650f5fbaf91)

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

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

~~~
shawn
My mistake. Thanks!

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FPGAhacker
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.

~~~
FPGAhacker
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.

~~~
dvfjsdhgfv
Don't hesitate to do that!

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Patient0
"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.

~~~
foxes
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.

~~~
Patient0
The history seems to be summarized well in this wikipedia entry:
[https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of...](https://en.wikipedia.org/wiki/\(%CE%B5,_%CE%B4\)-definition_of_limit)

------
foxes
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.

~~~
pedrosorio
> its called the Euler method

Disclaimer: and it sucks

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tmpmov
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.

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koalala
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.

------
z2
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.

~~~
SOLAR_FIELDS
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.

------
mayankkaizen
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.

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crispyambulance
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.

~~~
egonschiele
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!

~~~
1ba9115454
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?

~~~
egonschiele
The Procreate app on the ipad

~~~
shawn
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!

~~~
egonschiele
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](https://streak.club/s/8/daily-art) to keep motivated.

~~~
shawn
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](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!

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

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billfruit
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.

~~~
bsznjyewgd
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...](https://en.wikipedia.org/wiki/Notation_for_differentiation#Euler's_notation)

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toolslive
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?

~~~
tinco
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.

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

What do you mean?

~~~
tinco
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.

~~~
syntheticcdo
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.

~~~
tinco
Ah cool, thanks!

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

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

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stevewilhelm
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.

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

