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Simplification Problem before Finding Derivative

  1. Aug 8, 2012 #1
    1. The problem statement, all variables and given/known data

    I have a bit of confusion surrounding one of my basic derivative problems:

    Find the derivative of the following:

    \begin{equation}
    f(x) = \frac{x - 3x{\sqrt{x}}}{\sqrt{x}}
    \end{equation}

    2. Relevant equations

    None, I believe.

    3. The attempt at a solution

    I think I can simplify this, so I separate them:
    \begin{equation}
    \frac{x}{x^{1/2}} - \frac{3x. x^{1/2}}{x^{1/2}}
    \end{equation}
    Using the indices laws, I add and subtract:
    \begin{equation}
    x^{-1/2} - 3x
    \end{equation}
    Now it's simplified as far as I can see, so I take the derivative:
    \begin{equation}
    \frac{-1}{2}x^{-3/2} - 3
    \end{equation}
    And my final answer is:
    \begin{equation}
    \frac{-1}{2\sqrt{x^3}} - 3
    \end{equation}
    I'm confused about simplifying the exponents the way I did. If I have to use the quotient law, I'm not too sure how to apply it correctly in this case.
     
  2. jcsd
  3. Aug 8, 2012 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Why do you claim that [tex] \frac{x}{\sqrt{x}} = \frac{1}{\sqrt{x}}?[/tex] The claim is false. If, as you claim, you are just adding/subtracting exponents, then you seem to be saying that 1 - 1/2 = -1/2.

    RGV
     
  4. Aug 8, 2012 #3
    Sorry. I made an error. Let me try fix it.
    When I simplify f(x) (hopefully correctly this time), I get:
    \begin{equation}
    x^{1/2} - 3x
    \end{equation}
    Then when taking the derivative, I get:
    \begin{equation}
    \frac{x^{-1/2}}{2} - 3
    \end{equation}
    My final answer is:
    \begin{equation}
    \frac{1}{2\sqrt{x}} - 3
    \end{equation}
     
    Last edited: Aug 8, 2012
  5. Aug 8, 2012 #4
    Writing your problem in terms of exponents instead of radicals make it easier:
    [tex]\frac{d}{dx}\frac{x-3x^{3/2}}{x^{1/2}}[/tex]
    Split that up to get
    [tex]\frac{d}{dx}x^{1/2}-3\frac{d}{dx}x[/tex]
    Then your answer is correct. To check your work, it is always good to integrate what you got:
    [tex]\int \frac{1}{2\sqrt{x}}-3\,dx=\frac{1}{2}\int x^{-1/2}\,dx - 3x=\sqrt{x}-3x[/tex]
    which is what we started with.
     
  6. Aug 8, 2012 #5
    I haven't started on integration yet, but I do know I'll be doing it soon. Thanks a lot for all your patience and time Millenial and Ray Vickson.
     
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