Derivatives

So my Calculus AP teacher has jury duty this week. Being the teacher she is, she is still having us take our weekly quiz on whatever we went over that week even though she won't be at school to teach this week. She left us our work and the sub with instructions to hand out the quiz on Friday. We need to know how to use the formal definition of a derivative and the power rule. I can do the power rule fairly easily, but the formal definition trips me up at times, specifically with this problem (and others like it):

f(x) = 2(sqrt(x))

Using the power rule I got f'(x) = 1/sqrt(x), and assuming this is correct, I have no clue how to go about reproducing this using the formal definition. The problem specifically asked we solve it formally, but as I was stuck I used the power rule to find an ending value to work towards but am stuck at attempting to use the conjugate.

f'(x) = [(2(sqrt(x + delta(x))) - 2(sqrt(x))) / delta(x))] * [(2(sqrt(x + delta(x))) + 2(sqrt(x))) / (2(sqrt(x + delta(x))) + 2(sqrt(x)))]

The only way I know to deal with fractions with square roots is using a conjugate (math teacher last year didn't teach us a thing, as he figured we were all failures anyway leaving those of us planning to go into AP screwed) but it isn't working out so well lately.

Any ideas/help would be appreciated.

wow I'm dumb I can't even read the question correctly

disregard

http://i.imgur.com/TrsRd.jpg
Even though I'm late, I spent 5 minutes writing this down.

Thanks. I guess I had the right idea but felt like I was looking at a bunch of gibberish when I attempted it. It may have been the way I wrote it seeing as it suddenly looks much clearer than before. The only other thing troubling me so far would be fractions. I still have no clue what delta(x) cancels on the bottom of the formal definition when dealing with f(x) = x / (x - 1). This was an example in the book. Not sure if it's the denominator or the numerator but it just disappears in the book with no explanation as to how.

delta x in the denominator cancels out delta x in the numerator. In order to obtain a derivative using its definition, you must simplify the numerator to the point where it's just a number * delta x, allowing the delta x in the denominator to be cancelled out. Think of it as simplifying a fraction: for example, in order for 2/6 to be simplified, one must divide by the LCM (2). In the case of the definition of a derivative, the LCM is delta x.

Okay, that makes far more sense than the book did. I'll see if anything else is bothering me after school tomorrow. My teacher said she'd try to get another teacher to fill in for her tutoring sessions after school on Thursdays, so I'll decide what I need to know then, if anything.