Why is calculus so ... EASY ?

Calculus made easy, the Mathologer way :)

00:00 Intro
00:49 Calculus made easy. Silvanus P. Thompson comes alive
03:12 Part 1: Car calculus
12:05 Part 2: Differential calculus, elementary functions
19:08 Part 3: Integral calculus
27:21 Part 4: Leibniz magic notation
30:02 Animations: product rule
31:43 quotient rule
32:18 powers of x
33:10 sum rule
33:52 chain rule
34:54 exponential functions
35:30 natural logarithm
35:56 sine
36:32 Leibniz notation in action
36:43 Creepy animations of Thompson and Leibniz
37:00 Thank you!

Online version of Silvanus P. Thompson's book "Calculus made easy" at Project Gutenberg:

There is also a version of this book annotated by the great Martin Gardner. That's the one to get if you after a hardcopy.

Paranormal distribution maths t-shirt:

Music: Morning mandolin by Chris Haugen and Game changer by ikoliks.

Thank you very much to Eduardo Ochs for his subtitles in Brazilian Portuguese.

Burkard

[joinkusbelinkiusthethird]

Looking back on calculus, most of the things I actually had issues with were not the core concepts, but in fact was my ability to perform algebra without making small mistakes, remembering and applying trigonometric identities, and getting used to new notation. To anyone going through Calculus I urge you not to stress too much about it, just do your best it comes in time!

[Adomas_B]

Calculus is incredibly easy and trivial if you already know calculus

[natalieeuley]

The concepts of calculus are easy, and so is making programs to do it. The hard part of calculus is how they teach it in school. They want you to solve it with all of the rules to memorize. But memorizing all those rules- and the exact situations in which to use them- is the difficult part. The ideas of differentiation and integration can frankly be understood by anyone who can understand the area of a circle and how to graph a line; in other words, a late elementary school student or older. But for me, calculus was the first math class where suddenly there was no ability to look at a problem and know immediately how to solve it; you had to try different things on the same problem until it worked. And that does make it more difficult than any previous math class. Granted, it really doesn't have to be that way. Teachers could teach it differently and you wouldn't have that problem.

[bobbwc]

"Calculus is easy, if you are me." - Gottfried Wilhelm Leibniz

[dirtymike]

It is easy. The harder part is learning all the prerequisite material you need to know to start to learn calculus. But if you know algebra, trig, and geometry really well, calculus is incredibly easy.

[delduq]

One of the first exams I had in physics the teacher gave us velocity over time graphs and we had to “be the car” and move in distance over time. Now that I’m 63 and still remember this tells me it was one of the best learning tasks I ever had.

[jjreddick]

Calculus makes things easier once you know it. Learning integration is a perfect example. First we were taught to integrate using infinite rectangles, trapezoids, etc. It was tricky to find the correct formula and take the limit. However, once we were taught anti derivatives, it became much easier.

[maad]

You know Calculus is difficult when someone writes a whole book about how easy it is.

[EebstertheGreat]

In high school, I was surprised by how by far the hardest parts of Calc I and II simply involved a lot of steps of algebra. Things like partial fraction decomposition are a major pain, but actually integrating the resulting rational functions was very straightforward--once you did the necessary algebra (completing the square, etc.). Then in Calc III, I found it was much the same. Vector algebra is obnoxious, but the calc part really isn't so bad. I will say though that it gets much worse. Nonlinear differential equations are way harder than anything you have to deal with in a high school algebra class. I'd sooner factor ten solvable quintics than stare at a system of nonlinear PDEs until my brain melts.

[shawnlove]

I found calculus to be really easy when I first learned it, but it was always the algebra that held me back. Just as they say, people take calculus to finally fail algebra.

[morehmathematics]

Silvanus Thompson’s book “Calculus Made Easy” sparked my interest in higher math when I was younger and definitely influenced me into becoming a math major, absolute gem of a book every calc student needs a copy

[mrgeorgejetson]

Awesome video. I learned late in life that this kind of math isn't something I'm naturally bad at, just something that requires more effort on my part than, say, writing an essay on Wittgenstein's late period thought. But then again, calculus is something that requires a lot of effort for MOST people. Anyway, it's great to have resources like this, which are obviously the product of a great deal of passionate labor on the part of Mathologer.

[johnopalko]

When you mentioned _Calculus Made Easy_ I thought, "Hey, I have that book!" and ran to the bookshelf to retrieve it. As it turns out, no I don't. I have a book called _Calculus the Easy Way_ by Douglas Downing of Yale University, © 1982.

It's a fun little book wherein the protagonist is involved in a shipwreck and washes ashore in the land of Carmorra where he, in essence, helps its denizens invent calculus in order to answer burning questions involving the speeds of trains, the areas of fields, the simple harmonic motion of a spring-powered chicken scaring machine, etc.

[evanbarkman]

I always liked to describe differentiation as just a bunch of rules you have to apply and it's usually straight forward how to do it.
Integration on the other hand consists of either knowing the answer or trying to manipulate the function until you do.... with the optional third step of giving up and looking it up on a table.

Also I like that the music got way more epic as soon as you got to the chain rule.

[thomaskember]

When I started learning Calculus in High School, I began to realise that everything I had learned in maths before then from simple arithmetic, geometry, algebra and trigonometry was leading up to it. Does this mean that one reason we learn how to add and subtract is so that we can eventually do Calculus?

[waqarbaig]

I agree. It is also a very "attractive" type of math. What I mean is that once you get it over with, by completing the classes, you kinda want to go back and continue doing calculus. Its not the same with linear algebra or other maths. At least thats my opinion on cal.

[jimmy]

Yet again, you manage in 30 mins to better explain something than my maths teachers could over a year. Bravo, sir!

[matthewcerini]

Never commented here before... Burkard, you seem like the coolest person! Loved every one of your videos that I have watched. I wish I had the internet when I was a kid. Learning math with you as a kid would have been so much simpler and so much more fun. Thanks for everything! You rock!

[JRRodriguez-nupo]

65yo, haven't used calculus in over 40 years, still remember the heart of calculus and find it easy to follow this wonderful presentation. I see calculus as "depowering" or "powering" operations. Exponents become addition and back again. I imagine this ranking of increasingly more powerful operations, and calculus as the rules for going up and down in effect.
Very similar to the way explained here by graphs and the table of elementary operations. Go up for slope, go down for area. From the first it seemed so intuitive back when I was young.
Then again, so did dancing molecules which led to my career in medicine and biochemistry as an expert in single carbon metabolism ( the dance of the B vitamins).
Mathologer is a wonderful channel because he loves this. There's an inherent beauty that simplicity brings; but you must first love knowing for the sake if knowing, not some other goal.
Sorry, an old man reminiscing of when he still had a functional mind here. Soon again, I will think.

[pythagorasaurusrex]

I can't recall what famous mathematician once had this quote (in German): "Ableiten ist Handwerk, aber Integrieren ist eine Kunst".
In english like: "Taking a derivate is a craft, but integration is art."
When you know the rules, you can take the derivative of any function, no matter how complicate it is.
But integration can be a pain in the butt. Without the help of substitution tables, I was quite busted during my studies at university when it came to quotient of functions.
Thanks Burkard for this video (as always)!