Reddit mentions of Gravity

I don't say this to be discouraging: Most people don't really have any idea what doing Physics at a high level looks like. I decided in High School that I wanted to be a physicist, and as luck would have it I'm a graduate student and I still enjoy it, but truth be told, the exposure you have in High School doesn't really prepare you for the reality. All that to say: There's no reason to decide at thirteen years old that you need a PhD in Physics! Maybe once you learn math beyond trig you'll decide it isn't for you, or maybe you'll love math and want to switch to a math degree.

All right, now that that's out of the way... You said you're learning trig, that's good, you need it. You also need some basic algebra skills. Then try to teach yourself basic calculus (limits, derivatives, integrals). Then you want to learn Linear Algebra and at least Ordinary Differential Equations.

You can also do some basic physics reading before you've learned the essentials. I really like George Gamow's books for this - he was a very well know and important physicist who also happened to write very accessible books that are very much for lay people but that also don't shy away completely from the math. I really enjoyed this one in particular.

For mathematics, I love Dover books - they're cheap AND good. Shilov, I've found, is clear and readable. This might not be introductory level, but it's inexpensive and let's you see what you're getting yourself into.

Last bit of advice for Physics is what one of my old high school teachers used to say - draw, label, and you can't go wrong. It's still mostly true.

to me, the most interesting mathematical history books are those with a targeted goal, meaning they cover the historical aspect of a specific topic, time period, person or group of people, etc. these, at least in my opinion, provide the most insight, as some of the more general books, particular those written for the general public, are too overarching to get much out of. also, another suggestion is to find biographies of mathematicians or physicists. i personally do not find any math history prior to the 1500-1600s or so that interesting, so all of my suggestions are appropriately biased.

a few suggestions are:

• a history of vector analysis by michael j. crowe
  
• isaac newton by james gleick
  
• isaac newton by gale e. christianson
  
• the clockwork universe: isaac newton, the royal society, and the birth of the modern world
  
• gravity by george gamow
  
• one two three ... infinity by george gamow
  
• the mactutor history of mathematics archive - the biographies index is really pretty good and is often a starting point for me.
  
• ams notices - great for finding expository articles and interviews
  
  i don't know of many books that follow the history of a certain subject that are any good. most of my historical knowledge regarding a certain subject is taken from multiple sources. a good method is to read articles, track down the books and/or papers that are referenced by a certain book or paper you like, find biographies of the mathematicians involved, etc. for instance, interviews with mathematicians (like those in the ams notices and the one i posted the other day) are fantastic troves of information regarding the history of certain topics, because the story is being told by a direct participant.

If f(x) = x^5 , then...

f'(x) = 5x^4 (first derivative of f(x), with respect to x)

f''(x) = 20x^3 (second derivative of f(x), with respect to x)

f'''(x) = 60x^2

f''''(x) = 120x^1 = 120x

f'''''(x) = 120x^0 = 120

f''''''(x) = 0

In case you didn't see the pattern, if f(x) = ax^b then f'(x) = abx^(b-1) . This is detailed here, under "derivatives of powers" under "derivatives of elementary functions":
http://en.wikipedia.org/wiki/Derivative#Derivatives_of_elementary_functions

This trick only works for polynomials. If you want to take derivatives of more complex functions, like f(x)=x! or f(x)=sin(x) or f(x)=e^x*, that is more difficult.

If you are interested in this stuff, there is a book I would recommend. I, too, learned about derivatives years before I took high school calculus. But I didn't stumble on to it myself, as you did. I wasn't quite that smart, lol. Somebody gave me this book, called "Gravity" by George Gamow. Its a really great little book. It introduces basic calculus, in the context of basic physics (velocity, acceleration, etc). It goes through exactly the sort of stuff you are asking about, and explains it better than I can.

Here's the amazon link. Its not available on Kindle as an ebook, but maybe some other place sells it as an ebook, I don't know. Otherwise, I highly recommend buying a printed copy. It goes through exactly the sort of stuff you are asking about, and explains it better than I can:

http://www.amazon.com/gp/product/0486425630/ref=kinw_rke_tl_1

I think you are on the right track. Take Physics C AP and Calculus BC your senior year and maybe continue taking programming. These classes will help you get ahead of your peers in College, especially Physics C which covers a great deal of material for a high school physics course. Also, apart from taking the right curriculum, I think the most important aspect of majoring in any field is having an interest in it. If your high school offers scientific research, like my high school did, I would enroll in the class. If you are lucky, you may be able to do some research in a nearby college, something that will definitely boost your college app and give you important experience. Also, I have some physics book recommendations that I highly recommend that you read at your age.

Check out these three books written by George Gamow. He has a talent in explaining difficult physics concepts to those who may not have that advanced of a scientific background.
One Two Three...Infinity, Thirty Years That Shook Physics,
Gravity