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Math Help - Need help with these derivatives

  1. #1
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    Need help with these derivatives

    , show that ; for this problem I got






    , show that ; for this problem I got




    , show that ; for this problem I got



    , show that ; for this problem I got




    , show that ; for this problem I got



    I know how to get the first derivatives for all these problems, but I'm having problem getting to these alternate derivative forms. Please help. Thanks.
    Last edited by rainbowbaconz; November 5th 2012 at 07:47 PM.
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  2. #2
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    Re: Need help with these derivatives

    Then sow us what you got please...
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    Re: Need help with these derivatives

    Sorry. I edited the post and included the solutions that I have so far.
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    Re: Need help with these derivatives

    Hello, rainbowbaconz!

    f(x) \:=\:\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^3

    \text{Show that: }\:f'(x) \:=\:\frac{3x^3+3x^2-3x-3}{2x^2\sqrt{x}} . What a STUPID way to write the answer!

    We have: . f(x) \:=\:\left(x^{\frac{1}{2}} + x^{-\frac{1}{2}}\right)^3

    Then: . f'(x) \;=\;3\left(x^{\frac{1}{2}} + x^{-\frac{1}{2}}\right)^2\left(\tfrac{1}{2}x^{-\frac{1}{2}} - \tfrac{1}{2}x^{-\frac{3}{2}}\right)

    . . . . . f'(x) \;=\; 3\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2\cdot\tfrac{1}{2}\left( \frac{1}{x^{\frac{1}{2}}} - \frac{1}{x^{\frac{3}{2}}}\right)

    . . . . . f'(x) \;=\;\frac{3}{2}\left(\frac{x+1}{\sqrt{x}}\right)^  2\left(\frac{x-1}{x^{\frac{3}{2}}}\right)

    . . . . . f'(x) \;=\;\frac{3}{2}\cdot\frac{(x+1)^2}{x}\cdot\frac{x-1}{x^{\frac{3}{2}}}

    . . . . . f'(x) \;=\;\frac{3(x+1)^2(x-1)}{2x^{\frac{5}{2}}}
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    Re: Need help with these derivatives

    Thanks for the reply Soroban. I understand the process, but how would you go about getting to the alternative form that I posted for that problem?
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