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Thread: How do I differentiate this equation?

  1. #1
    Member
    Joined
    Oct 2010
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    How do I differentiate this equation?

    I kept having problems trying to differentiate the following equation with respect to h:

    $\displaystyle
    \frac{d}{dh} (\frac{100}{h} + 36h(\frac{5}{h})^{1/2})
    $

    What I did was I tried to split it into two:
    $\displaystyle
    \frac{d}{dh} \frac{100}{h} + \frac{d}{dx}36h(\frac{5}{h})^{1/2}
    $

    then..

    $\displaystyle
    \frac{-100}{h^2} + \frac{d}{dx}36h(\frac{5}{h})^{1/2}
    $

    from here, I use product rule on the second term:
    $\displaystyle
    \frac{d}{dh}36h(\frac{5}{h})^{1/2}
    $
    $\displaystyle
    = \frac{d}{dh}(36h) (\frac{5}{h})^{\frac{1}{2}} + \frac{d}{dx} (\frac{5}{h})^{\frac{1}{2}} (36h)
    $
    $\displaystyle
    = 36(\frac{5}{h})^{\frac{1}{2}} + \frac{1}{2}(36h)(\frac{-5}{h^2})(\frac{h}{5})^{\frac{1}{2}}
    $
    $\displaystyle
    = 36(\frac{5}{h})^{\frac{1}{2}} - \frac{90}{h}(\frac{h}{5})^{\frac{1}{2}}
    $

    the final result of the differentiated equation that I got is..
    $\displaystyle
    \frac{-100}{h^2} + 36(\frac{5}{h})^{\frac{1}{2}} - \frac{90}{h}(\frac{h}{5})^{\frac{1}{2}}
    $

    which is wrong!

    And I don't know what is wrong and how I should do it.
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  2. #2
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
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    Quote Originally Posted by xEnOn View Post
    I kept having problems trying to differentiate the following equation with respect to h:

    $\displaystyle
    \frac{d}{dh} (\frac{100}{h} + 36h(\frac{5}{h})^{1/2})
    $

    What I did was I tried to split it into two:

    $\displaystyle
    \frac{d}{dh} \frac{100}{h} + \frac{d}{dh}\left(36h(\frac{5}{h})^{1/2}\right)
    $

    then..

    $\displaystyle
    \frac{-100}{h^2} + \frac{d}{dh}\left(36h(\frac{5}{h})^{1/2}\right)
    $

    from here, I use product rule on the second term:

    $\displaystyle
    \frac{d}{dh}\left(36h(\frac{5}{h})^{1/2}\right)
    $

    $\displaystyle
    = \left[\frac{d}{dh}(36h)\right] (\frac{5}{h})^{\frac{1}{2}} + \left[\frac{d}{dh} (\frac{5}{h})^{\frac{1}{2}}\right] (36h)
    $

    $\displaystyle
    = 36(\frac{5}{h})^{\frac{1}{2}} + \frac{1}{2}(36h)(\frac{-5}{h^2})(\frac{h}{5})^{\frac{1}{2}}
    $

    $\displaystyle
    = 36(\frac{5}{h})^{\frac{1}{2}} - \frac{90}{h}(\frac{h}{5})^{\frac{1}{2}}
    $

    the final result of the differentiated equation that I got is..

    $\displaystyle
    \frac{-100}{h^2} + 36(\frac{5}{h})^{\frac{1}{2}} - \frac{90}{h}(\frac{h}{5})^{\frac{1}{2}}
    $

    which is wrong!

    And I don't know what is wrong and how I should do it.
    $\displaystyle \displaystyle\left(\frac{5}{h}\right)^{\frac{1}{2} }=\frac{\sqrt{5}}{\sqrt{h}}$

    $\displaystyle \displaystyle\sqrt{\frac{5}{h}}\frac{d}{dh}36h+36h \frac{d}{dh}\left(\frac{5}{h}\right)^{\frac{1}{2}} =36\sqrt{\frac{5}{h}}+36h\sqrt{5}\frac{d}{dh}h^{-\frac{1}{2}}$

    $\displaystyle =\displaystyle\ 36\left(\frac{5}{h}\right)^{\frac{1}{2}}+36h(5)^{\ frac{1}{2}}\left[-\frac{1}{2}h^{-\frac{3}{2}}\right]=36\left(\frac{5}{h}\right)^{\frac{1}{2}}-\frac{36h(5)^{\frac{1}{2}}}{2h^{\frac{3}{2}}}$

    $\displaystyle =\displaystyle\ 36\left(\frac{5}{h}\right)^{\frac{1}{2}}-18\left(\frac{5}{h}\right)^{\frac{1}{2}}=18\left(\ frac{5}{h}\right)^{\frac{1}{2}}$

    which is what you have, if you simplify...

    $\displaystyle \displaystyle\ -\frac{90}{h}\frac{\sqrt{h}}{\sqrt{5}}=-\frac{18(5)}{\sqrt{h}\sqrt{h}}\left(\frac{\sqrt{h} }{\sqrt{5}}\right)=-\frac{18\sqrt{5}\sqrt{5}\sqrt{h}}{\sqrt{5}\sqrt{h} \sqrt{h}}=-\frac{18\sqrt{5}}{\sqrt{h}}$


    To make things easier, you could write

    $\displaystyle \displaystyle\ 36h\frac{\sqrt{5}}{\sqrt{h}}=36\sqrt{5}\sqrt{h}$

    Then

    $\displaystyle \displaystyle\ 36\sqrt{5}\frac{d}{dh}h^{\frac{1}{2}}=36\sqrt{5}\f rac{1}{2}h^{-\frac{1}{2}}=18\frac{\sqrt{5}}{\sqrt{h}}$
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