High School AP Calculus Help Wanted!!!! Please!

[ChadMKIII]

Hey Guys, I hate having to do this, but I'm shoving my ego aside for now cuz I can not see how the book got this for the life of me. I'm trying to follow an example. I believe this involves basic differentiation: Product Rule, Chain Rule, and Implicit Differentiation. Hopefully you geniuses are familiar with the names of the first two. If not but you have some help to offer plz PM me and I'll explain those rules.

So, here's the problem.
Part 1. Differentiate:

2x * sqrt(r^2-x^2)

The example in the book immediately jumps to their answer for the derivative w/ no steps explained. Here's their answer:

[2 * sqrt(r^2-x^2)] - [(2x^2)/(sqrt(r^2-x^2))]

Part 2. Simplification. They jumped from that above to this:

(2(r^2-2x^2)) / (sqrt(r^2-x^2))

So: When I tried to work this out, here is what I got for the differentiation:

[2 * sqrt(r^2-x^2)] - [(x)/(2r * sqrt(r^2-x^2))]

I have no idea how they got their answer, and definitely no idea how they got to the simplified version. They did not explain any of it (its part of a bigger problem). If you could explain it to me that'd be great! Thanks guys!

 

[jmanbball]

The r in the problem is a constant, so you treat it as a number, not a variable that is part of the derivative.

this is how I derived it:
2x*(1/2)*(-2x)/sqrt(r^2-x^2) + 2*sqrt(r^2-x^2)

The 1/2 comes from the deriv of the sqrt, and the -2x is the deriv of the inside of the sqrt.

that simps into
[-2x^2/sqrt(r^2-x^2)] + 2*sqrt(r^2-x^2)
which is what the book had

then you put them over a common denominator
[-2x^2/sqrt(r^2-x^2)] + (2*sqrt(r^2-x^2)*sqrt(r^2-x^2))/sqrt(r^2-x^2)

the sqrt times itself becomes just (r^2-x^2), and then you put them together:

(-2x^2 + 2*(r^2-x^2))/sqrt(r^2-x^2)
which becomes

[(-2x^2) + (-2x^2) + 2*r^2]/sqrt(r^2-x^2)

then finally simplifying to

(-4x^2 + 2r^2)/sqrt(r^2-x^2)

The book factored out a 2 in their simplified solution

 

[SupraDerk]

haha, dah, you beat me to it

 

[Shytheed Dumas]

Calculus = (squirt*ftl)^nth degree

[calc is the sole reason besides excessive alcohol consumption that damn near got me thrown out of college]

 

[ChadMKIII]

Hahahaha, I just realized another few mistakes I was making, mostly forgetting stupid algebra rules and stuff . Got it now...

I thanked you in PM's but just wanted to do it publicly as well, thanks so much guys!