I’ve taken Pre-Calc, Calc 1, 2, and 3, as well as Advanced Calc. I know all the differentiation and integration rules backwards and forwards.
The thing I don’t quite get is why the rules work.
A basic example is the power rule:
If f(x) = x^n, then f’(x) = nx^(n-1).
Great, got that. But why is this so?
Sheesh, that should read Explain to ME…"
Could a mod kindly fix the mistake? Thanks!
The basic deinition of the derivative f’(x) is:
f’(x) = lim {[f(x+h) - f(x)]/h} as h -> 0
Basically you write each function f(x) using the argument x+h and x, subtract, and divide out one h. Then set h = 0 (if there is a problem with this–say h is in a denominator somewhere still–then you either did it wrong or the derivative does not exist).
For your example, f(x) = x^n. Therefore f(x+h) = x^n + nx^(n-1)h + …, where “…” is a bunch of terms that have h at least to the power of two (if this doesn’t look familiar, review the binomial theorem). Subtracting leaves nx^(n-1)h + …, and dividing by h leaves nx^(n-1) +… where “…” is now a bunch of terms that include h. Set h=0 and you’ve got f’(x) = nx^(n-1).
That’s just one example, and there are a few points I’ve glossed over (e.g. the binomial theorem is usually proved only for integer values of n, but the formula is true for any value of n). But the differentiation “rules” work because the can be proved as a general case of this basic definition. Integration is just the reverse of differentiation, so whenever you find a derivative “rule” you’ve found a corresponding integration rule. A quick and dirty answer, but it’s really all you need 
It all makes so much more sense now, thank you! I reread my old calc 1 book and the definition you supplied, tried a few probs, and poof! lightbulb moment.
I heart calculus!
DC
aka Math Geek Extraordinaire
Speaking of calculus, what’s a good place to start learning it? I didn’t have to take any Math in college past the minimum requirements for the AA degree.
If you’re looking for books to help you learn on your own, I would recommend starting with one that gives you a thorough undstanding of pre-calculus concepts and trigonometry.
For Calc 1 - Calc 3, I used this book and highly recommend it. They work through the concepts and several examples in a way that makes sense, even without a teacher always handy to explain things. The solutions manual should be available online as well.
DC
For d/dx (sin x) = cos x, we did a proof in my first year of Calculus that involved some kind of clever drawing of triangles and circles. A quick google search gives me this much more algebraic and less geometric proof.
One way to deal with the relationship between ln(x) and 1/x is to simply define ln(x) as the antiderivative of 1/x, and then define e to be the number such that if ln(x)=y then e^y=x. From there, you can “prove” the definition of e that you probably learned in pre-calculus, with all those compound interest problems.
What else is there? d/dx(tan x) uses the quotient rule, which you can also prove by setting up the formula in CJJ*'s post… There must be other strange functions I’m not thinking of, but I hope that mostly answers your question.