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Derivatives of Rational Powers

  1. Oct 28, 2008 #1
    Hello!

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

    The problem statement, all variables and given/known data
    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
     
  2. jcsd
  3. Oct 28, 2008 #2

    statdad

    User Avatar
    Homework Helper

    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

    [tex]
    (x^2 + 2)^{1/3} (x^2-5x+1)^2 + (x^2 + 2)^{4/3} (x^2 - 5x + 1)
    [/tex]

    you would factor out

    [tex]
    (x^2+2)^{1/3} (x^2 - 5x + 1)
    [/tex]

    from both terms in the sum, and obtain

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

    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.
     
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