I'm away for like two months and people start messing around with Calculus. Crap.
So in case anyone else has this curiousity:
I do not understand why, for problems like e^(2x) or 5^(3x), the exponent is considered the "inner function", whereas with problems like (x^2 + 4)^3, the inner function is considered x^2+4 and the outer function is the exponent. Is it because for (x^2 + 4)^3 the exponent is applied to the inner function whereas e^(2x) is interpreted as (e^(2x))?
It may very well be that I learned this and have long forgotten it, but I cannot for the life of me remember any instance of this whole inner function and outer function thing. It may be due to the different curricula. We probably called it by other names.
There's a decent-ish explanation here: http://oregonstate.edu/instruct/mth251/ … nRule.html . This isn't really a problem in Calculus per se.
I do use the chain rule given at the end of the article (and certainly use composition and decomposition), which suggests that I probably learned this stuff, but fuck me if I remember "inner" and "outer" or the pointless drilling related to it.
The "inner" function, of course, is always the one that has x in it (oh, but see below), because it's the g in f(g(x)). F is a function of the "expression" g(x). So, for instance, e^(2x) is the function f(u) = e^u applied to g(x) = 2x (which suggests that 2x is the "inner" function). (x^2 + 3)^2 is the function f(u) = u^2 applied to the function g(x) = x^2 + 3, which suggests that x^2 + 3 is the "inner" function (i.e. it's the one "in" f0.
As for this one:
Of course, there's shit like d/dx sine(x)^(3x).
You were right about the track you chose, see http://www.analyzemath.com/calculus/Dif … ative.html for related example. It's a rather pointless exercise in symbolic manipulation. The good news is that functions like sin(x)^(3x) (or anything of the form f(x)^g(x)) don't really occur in real-life processes. I don't think I've ever had to do something like that in more than a decade now. The technique itself a useful trick sometimes, but nothing else.
Don't sweat *too* hard. Especially if you want to program computers for a living, you won't have to take the derivative of anything more complex than a polynomial, simple sin/cos and/or power functions for a long, long time, and when you *do* have to take a more complex one, it will be perfectly acceptable to ask a real mathematician.
THE NOTE BELOW: After you'll be done with high school, where everything is dumbed down in the form of an endless lesson in taxonomy, terminology, fancy names and gazillion of rules, you'll get to university-level Calculus, where everything suddenly gets simpler for some reason, despite the actual subject being a lot more complex. One of the fancy things that you'll learn is how to take the derivative of functions that depend of *several* variables (e.g. 2 variables, which describe not a curve, but a surface). So you shouldn't take the "has x in it" as anything more than a rule of thumb.
Studies show that learning exotic programming languages like Haskell, LISP/Scheme accelerates neckbeard & facial hair growth.