I am a math-y sort of person, although not a professional math-er, and EVERY SINGLE TIME I have to re-remember "ok, which one is the negative exponent, and which one is the fractional exponent?"

I wish fractional exponents were taught instead of ever seeing radicals, and I wish negative exponents were taught alongside learning fractions, from the ground up.

This is great, I love derivative shortcuts and wish I'd had something like this 38, 52, and 55 years ago the 3 different times I took Calc 1. Also for the f' vs df/dx explanation, never knew that. I look forward to the lesson on integration, which I found/find much more difficult.

I just re-read my posting and I realize (after decades!) that a lot of the problem is that the concept "fraction" is in "fractional exponent", and also in the answer to "what does a negative exponent mean?"

There are quick and simple explanations as to why the exponent rules work like they do, but I'm almost out of reception for the afternoon! I'll write it up though, because it's quick and fun.

You can keep your exponents on the ground or they'll get bugs. That's why they are superscript.

I'd forgotten that explanation for the product rule. That is genuinely helpful. Good, clear writing, too.

You're welcome. Keep the product off the floor.

My recollection was that Newton's notation was the dot above the variable denoting derivative (usually with respect to time) - used sometimes in physics these days but not so much in math. Wikipedia clams that f'(x) is due to Euler and Lagrange.

I am very sure that you're right - I am now remembering the dot notation - and yet I'm also very sure I've read/heard the wrong version that I parroted.

Also, on my phone, 3/4 of the single term derivative shortcuts are missing from the chart, and the 3rd column of the operations chart is cut off. Oh well.

Here's an example of a source saying that f'(x) is newton's notation: https://books.physics.oregonstate.edu/GSF/leibniz.html

e really is a great number. I'd forgotten about how it was unique in derivatives too.

π is a shitty number in comparison.

And don't get me started on 9.

I first heard this in high school (copied this version from Reddit):
e^x and a constant are walking down the street together when the constant sees a differential operator coming their way. He starts to run away, and e^x asks "Why are you running away?" The constant answers, "That's a differential operator. If it acts on me, I'll disappear." e^x says "I'm e^x, I don't have anything to worry about," and keeps walking. When he reaches the differential operator, he says "Hi, I'm e^x."
The differential operator responds, "Hi, I'm d/dy."

r dr r.

I feel partially responsible for that.

I still remember the day my dad explained to me Euler's formula for expressing complex numbers in terms of e; blew my mind that it worked that way and I'm not sure it ever got unblown.

Also ha ha cis x, I didn't know that one but of course cisness is complex.

For 1 what you want is the "multiplicative numberline" in your head. 1 is the middle (so it's the zeroth power) and going to the right you have evenly spaced 10, 100, 1000, etc. and to the left .1, .01, .001, etc. Multiplying by 10 moves you one to the right, dividing by 10 one to the left, etc.

10^-2 is going left twice, 10^1/2 is going halfway from 1 to 10.

What's supposed to just be completely ingrained here is that 10^1/2 is between 1 and 10, while 10^-1 is smaller than 1.

That's weird I just wrote a long comment and it completely disappeared.

If your comment was a constant and derivative, it goes to zero.

I get the statements in 16 but I don't see how they amount to a punchline. Is that the joke?

Ok, here's my quick explanation on how to remember fractional and negative exponents.

24: Because d/dy(e^x) = 0, because it's a constant with respect to y.

26: Oh, ok. Explaining that actually made me laugh!

You're a cheap date.

Thanks for the exponent "fraction vs. negative thing"; when I say "I have to remember it each time" I mean that I go through basically that logic. But obviously I'm supposed to know it better than that.

I suspected that you were already reasoning it out for yourself - I almost put a disclaimer. But I still find these fun to write out.

The explanation in 25 is nice, but I really think you need some kind of log-scale numberline in your head to make it instant recall.

I don't have a log-scale numberline on quick retrieval in my head! I have to reason out log-scales each time.

This lesson exposed me as a fraud. Is it worth the trouble to try to remind myself about sin and cosin and tan, etc?

But I said I didn't expect the reader to remember those things!! My meta-post is a failure.

By the way, I appreciated 21, that's a useful visualization to keep in mind.

34:. So I don't need to understand any of this to continue with the lessons?

You really don't. You just need to trust that derivatives have quick shortcuts, and anything else I can explain when we need it.

37: Thanks, heebie!

Anybody else read Kevin Drum's synopsis of Godel's work? I liked it, and I learned something from it, about what Godel numbers are, and what Godel did with them, though I'm sure I couldn't explain it to satisfy people who know math.