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”