Turn Huh? to Aha! and Build Lasting Math Insight


thank you for releasing this - i have found this site endlessly useful. You put things in a way that just fits!

Its good to buy an ebook to take away with me and have things put into a more structured and familiar form (take as a compliment). Worth ever £ and i will be recommending to many!!

Kind Regards,


@Afsar: Thanks for the comment! I’m happy if anything can be used to help your students. Feedback is definitely welcome!

@Doug: You’re welcome, thanks for the comment and support! Yes, I want the ebook to be like a portable version of the site, I have some changes to the beta coming up that will help with just that :).

I have some decent intuition (beyond the proofs
) of the Chain and Power rules for derivatives. Just thought I’d share, whether or not you already know.

Chain: This “rule” is really the simple concept of “correcting” for different variables when you’re looking at relationships. It says that d/dx f(g(x)) = f’(g(x)) times g’(x)
In other words, figure out the direct relationship between f and g, not thinking about x just yet, then multiply by the relationship between g and x.
Example: Let’s say Al takes 3 steps whenever Bill takes 1 step. That would mean the ratio of Al to Bill is 3:1, and if Al were the y-axis and Bill were x, then there would be a line of slope 3/1. Now let’s say that Bill takes 2 steps whenever Carrie takes 1 step.
The question is, how many steps does Al take when Carrie takes 1 step?
Not hard. When Carrie takes 1 step, Bill takes 2 steps. And when Bill takes 2 steps, Al will have to take 6 steps!

Rate of A wrt B = 3
Rate of B wrt C = 2
Rate of A wrt C = (3)(2) = 6

Maybe it’s already obvious to you and everyone else, but judging by the misleading use of the word “rule” to describe this simple concept, I can’t be too sure. Tell me if I haven’t been clear and detailed enough.

Power: Geez, I wish I could draw a picture. This is a new insight, and I have only a visual, rather than arithmetical, intuition. It says that d/dx x^n = nx^(n-1)
Looks really weird, and the best proof I’ve seen uses the binomial theorem, but hopefully it’ll make more sense now, if it doesn’t already. (You know so much!)
Imagine a square, x^2, to start. Imagine the side of the square grow a tiny bit, which will be “dx”. How much does the area of the square increase? Well, there are two parts to this; two rectangles are formed, each with a long side of x (the side of the square) and a short side of dx, along with a super small square of side dx. Think (a+b)^2=a^2+2ab+b^2. The key point here is that dx^2 so small, you don’t even have to think about it. So the square grows by two thin rectangles with total area 2xdx. But the derivative is a ratio, which means when you compare the amount the area grows, 2xdx, to the amount the side grows, dx, you get 2x. Imagine two lines bulging out of the sides of the square! 2x!
Look at a cube, x^3. Using the same logic, imagine three squares bulging out of the faces of the cube! 3x^2!
This is the pattern: If x^n is your n-dimensional object, and x is the length of a side, then the area/volume of each bulging cross section is x^(n-1)dx. Actually, this has the same number of dimensions, except one of the sides is now the growth in x. How many bulges will we need? One for each dimension, n (visualize the cube).

You might have figured this out yourself by now, if this blog is any indication. Just wanted to share anyway. I’m a 16 yr old homeschooler, and after I first met you via your post on e a while back, I really got inspired to truly become comfortable with math, instead of just preparing for the ap tests! I was sort of already trying to understand, but you boosted and energized my efforts. Now I’m past the stage where you think math is a mess of logic, formulas, and conventions, and I still have my whole life ahead of me! Thanks.

Also, a typo I caught in your ebook:
Pg. 1: “Afterward” might be “afterwOrd”

@Kent: Thanks for the note! That’s really, really awesome that you’re starting to see math in this deeper way, I wish I had realized the importance when I was your age. I think you’ll find it helps you in a ton of subjects, not just math – looking for the deeper connections just helps everything click for me.

Chain rule: Yep, that’s a great way to look at it. Another way I’ve found is to think of it as a series of “wiggles”. That is, A moves 3x whenever you “wiggle” B, and B moves 2x whenever you wiggle C. So when you wiggle C just a bit, you get the multiplied effect you mention.

Power rule: That’s exactly how I think about it! For each dimension you have, you gain a “slab” on each side. So for x = 2 (square), you indeed gain 2 slabs, and so on.

It’s really great you’re finding these insights so early :slight_smile: [Thanks for the typo, it’s fixed now].

great ebook, i have purchase it hope to see more good sharing in the near future…keep it up!

@See Zeng Hoe: Thanks for the support!

when the final version will up?

@See Zeng Hoe: I’m pulling in the final round of feedback now, and aiming for late October for the final version.

UPDATE: Hi everyone, the final version is now available! If you have purchased a beta copy, you should receive the final version in your email. Please contact me if you didn’t receive it or are having trouble.

hi khalid,

Do you plan to publish a new ebook related to math?

@See Zeng Hoe: Yes, I’d love to publish another one – I’m thinking about what to include, and writing a few more posts :).

Hi Khalid, I love your website and wanted to buy the Beta version but due to some problems (i.e. my tendency to procrastinate and my absent-mindedness), I didn’t. And now it’s gone up to a price I don’t have a budget for at the moment… yikes! I’ll definitely buy it coz yr explanations on math concepts simply rock! is still saving money

I like a hard copy better tho, so if it’s out, pls announce it here, thanks!

@Ami: Hi, thanks for the comment & feedback! I’m thinking about lowering the price which should hopefully be in the range of more people :). And yep, I’ll definitely be announcing the hard copy as soon as it’s ready.

How much of this is articles from the website, other than the extra chapter?

@Andy: The extra chapter, foreward, and afterward are unique to the book (the premium version also includes the original powerpoint slides used to make the diagrams) . The other chapters are based on the articles, with changes/updates to make them flow nicely in a book format. The primary benefit of the book is having the content in a portable, DRM-free format for comfortable printing and reading. Hope this helps!

Cool, I’ll probably pick it up, I’d like to support the site anyway. By the way, you should make a Facebook fan page, I’m sure your readers would be glad to give you a little extra publicity.

@Andy: Thanks for the support, and great idea on the facebook fan page. I’m traveling now but will probably set one up when I return :).

I’m sad because I don’t have a credit card just cash. I think that if I had an organized ebook ot would be easier to understand because I would read in the right order. I can afford the ebook but don’t have a credit card :frowning:

@Seamus: I’ll see what I can do :).

@Kent hey man that’s amazing that you’re getting such “deep” insights already. Man seriously, this stuff is what should be taught in schools. These insights are the real deal of what math’s about.

I never got till today any intuitive understanding of the chain or power rule. I only knew the algebraic proofs… I didn’t even think an intuitive understanding was possible.