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Math Help - Derivatives...

  1. #1
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    Derivatives...

    Ugh I understand all my basic differentiation rules and my basic algebra but I get into a problems when I have longer problems like these. Sorry if it seems like I'm asking you to do the problem for me but would anyone mind giving me a little guidance?

    Differentiate and solve for T when dP/dT = 0 (A is a constant)

    <br />
P=2(\frac{A-\frac{1}{3}T^2}{\frac{2}{3}T})+\frac{2T}{3}+2\sqrt  {(\frac{T}{3})^{2}+T^{2}}<br />

    I just don't really know where to start or anything.. Thanks!
    Last edited by pakman134; March 22nd 2008 at 06:23 PM. Reason: Clarify
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  2. #2
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    the trick here is to use some algebra to simplify your expression. before doing any calculus

    P=2 \left(\frac{A-\frac{1}{3}T^2}{\frac{2}{3}T} \right)+\frac{2T}{3}+2\sqrt{\left( \frac{T}{3}\right )^{2}+T^{2}}

    \frac{4A}{3T} - T + \frac{2T}{3} + 2 \sqrt{T^2(\frac{1}{9} +1)}

    \frac{4A}{3T} -  \frac{T}{3} + 2T\frac{\sqrt{10}}{3}

    \frac{4A}{3T} + T\frac{2\sqrt{10}-1}{3}

    Now you differentiate.

    I am unsure about what "solve for T" means though.


    Edit: I noticed you defined what "solve for T" means, I am sure you can finish the problem from here, post if you have any problems
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  3. #3
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    Wow the power of simplification....... =p

    Got a little further and then I have problems again solving for T.. here's what i have so far....

    <br />
\frac{4A}{3T} + T\frac{2\sqrt{10}-1}{3}<br />

    dP/dT = (3A)(-1)(3T)^(-2)(3) + \frac{2\sqrt{10}-1}{3}

    0 = (3A)(-1)(3T)^(-2)(3) + \frac{2\sqrt{10}-1}{3}

    -\frac{2\sqrt{10}-1}{3}= (3A)(-3)(3T)^(-2)

    Not really sure how to take it any further or if my previous differentiation is even correct.... Would appreciate any help. Thanks!
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  4. #4
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    P= \frac{4A}{3T} + T\frac{2\sqrt{10}-1}{3}<br />

    \frac{dP}{dT} = \frac{-4A}{3T^2} + \frac{2\sqrt{10}-1}{3}

    if \frac{dP}{dT} = 0 we get

     \frac{4A}{3T^2} = \frac{2\sqrt{10}-1}{3}

    reciprocate and cancel the 3

     \frac{T^2}{4A} = \frac{1}{2\sqrt{10}-1}

    T = \pm \sqrt{\frac{4A}{2\sqrt{10}-1}}
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