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.