>> One formula appears to say the derivative is f(x) * g(x), and another 
>> appears to say it's f'(g(x)) * g'(x), which is different, but these 
>> are supposed to be different notations for the same formula...?
> 
> The first one is obviously wrong, unless you mixed up all the f's, g's 
> and little circles. It is difficult to tell without context.

Well, one formula says something like dy/dx = dy/du * du/dx, while the 
other says (f.g)'(x) = f(g(x)) * g(x).

>>> df(f(x))/dx = f'(f(x))*f'(x) = g(f(x))*g(x)
>>
>> Right. So basically, the function I'm looking for is f'(f(x))*f'(x)?
> 
> Yes.

Nice to know I got something right today...

>> And in general, the Nth composition will be ever more complex as N 
>> increases?
> 
> Yes.

Hmm.

>> ...what I *should* have done of course is
>>  http://www.wolframalpha.com/input/?i=derivative+f(f(x))
> 
> Except that it is the wrong answer, or rather the wrong question.

Yes. The query should be f[f[x]] instead of f(f(x)). If you ask for a 
higher iterate, it quickly becomes obvious that something is wrong.

> Just one more reason why it is better to learn the formulas yourself 
> instead of relying on a computer to help you all the time.

Oh, sure. But I find it useful to check against the computer. If the 
answers are the same, you probably did it right. If the answers are 
different, either you asked the computer the wrong thing or you stuffed 
up the math.

(For example, (Sqrt(3) a)^3 = 3 a^2. Or NOT, in fact - as Wolfram 
rightly pointed out...)

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