My own proof for the power rule in differential calculus · 11:44pm Feb 2nd, 2013
During long, boring lessons in high school, some kids whispered to each other, some doodled on the desktops or in their notebooks, some passed notes, others twirled their hair, chewed gum, or texted. Some actually listened to the teacher.
I was the one the who was doing calculus in my notebook. Don’t judge me.
I love proofs; if I understand a proof, or better, have some intuition, then I’ll never forget a formula. But I wasn’t content with the proof for the power rule in my textbook, as it assumed many things that were supposed to be “obvious,” but weren’t obvious to me. So, when we learned about the number e and the natural logarithm, I was fooling around with it to see what I could do. And I came up with my own proof for the power rule, which is exceptionally elegant. It uses logarithmic properties, e, and a bunch of other really cool stuff.
The power rule—probably one of the most important rules in differential calculus. It becomes so ingrained in your consciousness that, eventually, you’re not even conscious that you’re using it. And I proved it.
There are a few problems with my proof that I can see. One, it doesn’t work if the variable is equal to zero. Also, it presupposes the veracity of the chain rule—a rule that, still, to this day, despite having a very intuitive understanding of it, I still don’t understand the proof of. But we take what we can get, right?
So, here it is. And, because I’m unbelievably pretentious, I did the entire thing in Latin. A helpful user pointed out the grammatical flaws in my previous blog post, so, this is to see if I’ve gotten any better. It’s mostly numbers and equations, so even if you don’t understand Latin, you’ll still be able to understand the proof, hopefully. And if you do understand Latin, you can make fun of my grammatical errors, since making fun of someone’s second language is always humorous.
NOTE: This proof may exist elsewhere, but I independently discovered this proof—I swear:
Isn't this awesome? The power rule has just become! . . . has just become . . . a certain amount cooler than it is now. How much cooler? Well . . . it has become . . . well, on the function 1/x, the area under the curve between 1 and e^1.5 times cooler:
Yeah, that many more times cooler, is what I'm driving at, assuming the units under the curve are dimensionless.
I hope you liked that. That’s all I had to say today, really. Oh, and I have a new story.
EDIT: Holy shit, I derped hard. Coolness function is fixed now. The last one was an improper integral.
I had a headache before I read any of that.. Now I have a migraine..
Calculus I can handle. Calculus with latin proofs, no. I think I need to go to sleep..
... What the hell are you? You certainly aren't human, too smart.
LOL I was kinda that student too... Anyways i heartily approve but lack the time right now to look @ your proof, still seems awesome.
I barely remember a thing from my calc classes...
Ooh, calculus. I can't read Latin, but the proof seems to check out just fine. Power rule is indispensable in usefulness and much better than deriving though the limit of the difference quotient. That was a nightmare waiting to happen with higher degree functions.
This makes Trixie look forward to taking Calc next year.
Write a story in which Twilight Sparkle (or better yet, Pinkie Pie) explains calculus to Rainbow Dash. Within a week, the entire fandom will know calculus.