Derivatives of Rational Powers

[Alexstre]

Hello!

I've been solving a few of these problems but I'm stuck on this one, trying to simplify one of the steps.

Homework Statement
Find dy/dx of: y = x(x^2 +1)^1/2

Attempt at a solution
y1 = x (1/2)(x^2 + 1)^-1/2 * (2x) + (x^2 +1)^1/2 * 1

I get the the equation above but I have no clue how to simply it. I check the answer manual and found that the next step should be:
(x^2+1)^-1/2 * (x^2+x^2+1)

With that, I can simplify and solve the problem, I just don't know how to get there from my equation.

Could anyone point me in the right direction?

Thanks

 

[statdad]

A general 'rule' in these types of problems is this: to simplify them, factor out the highest power of common terms.

For an expression like this one

$$(x^2 + 2)^{1/3} (x^2-5x+1)^2 + (x^2 + 2)^{4/3} (x^2 - 5x + 1)$$

you would factor out

$$(x^2+2)^{1/3} (x^2 - 5x + 1)$$

from both terms in the sum, and obtain

$$(x^2 + 2)^{1/3} (x^2 - 5x + 1) \left((x^2 - 5x + 1) + (x^2 + 2)^{3/3} \right)$$

and then proceed to simplify the expression inside the final parentheses.

Look for common factors in the terms of your expression, and use this procedure. It's tempting to skip steps by doing them in your head: don't do this until you're a little
more familiar with this type of work.