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Math Help - derivatives with a twist

  1. #1
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    derivatives with a twist

    Hey, just found this forum and wanted to give a quick thanks beforehand.

    I took calc a year ago and did ok, but now i need to know some calc for an econ class im in.

    Here's a sample question...
    Q=10L^.75 K^.25
    In my calc class we just had to do the math, and i would multiply the exponent to the coefficient and minus the exponent by 1. I'm 99.9% this is correct.

    But my econ teacher wants us to caclucate to sets:
    a. calculate derivative Q/derivative L
    b. calculate derivative Q/derivative K

    I'm lost on that part. Thanks
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  2. #2
    Senior Member mollymcf2009's Avatar
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    Is the sample equation you posted the equation you are supposed to use to determine the two sets for a value? You will need to use the product rule to get the derivative for the sample problem because you are multiplying two variables and the quotient rule for the Q/K problems. Can you be more specific about what you are solving for?
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  3. #3
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    Sorry if i wasnt too clear.
    Heres the exact problem:

    Consider the following function Q= 10L^.75 K^.25
    a. Calculate derivative Q/derivative L
    b. Calculate derivative Q/derivative k

    I figure if i can figure out a or b, i can do the other one.
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  4. #4
    Senior Member mollymcf2009's Avatar
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    Product rule:
    Q = lk
    Q' = l'k +lk'

    So,

    Q = 10l^{.75}k^{.25}


    Q' = 7.5l^{-.25}k^{.25}  +  10l^{.75}.25k^{-.75}

    I laid out the product rule for you. See if you can finish it and then go from there. Come back if you get stuck!!
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  5. #5
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    Quote Originally Posted by foxracing View Post
    Sorry if i wasnt too clear.
    Heres the exact problem:

    Consider the following function Q= 10L^.75 K^.25
    a. Calculate derivative Q/derivative L
    b. Calculate derivative Q/derivative k

    I figure if i can figure out a or b, i can do the other one.
    The question as you've posted it requires the idea of a partial derivative. have you learned about this?

    To get \frac{\partial Q}{\partial L}, you treat K as if it were a constant. Then:

    \frac{\partial Q}{\partial L} = 0.75 L^{-0.25} K^{0.25}.

    The other is done in a similar way and is left for you to attempt.
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  6. #6
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    Quote Originally Posted by mollymcf2009 View Post
    Product rule:
    Q = lk
    Q' = l'k +lk'

    So,

    Q = 10l^{.75}k^{.25}


    Q' = 7.5l^{-.25}k^{.25} + 10l^{.75}.25k^{-.75}

    I laid out the product rule for you. See if you can finish it and then go from there. Come back if you get stuck!!
    The problem as posted requires the idea of a partial derivative. See my post above. So gettting a total derivative (with respect to x ....?) is not correct.
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  7. #7
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    Quote Originally Posted by foxracing View Post
    Hey, just found this forum and wanted to give a quick thanks beforehand.

    I took calc a year ago and did ok, but now i need to know some calc for an econ class im in.

    Here's a sample question...
    Q=10L^.75 K^.25
    In my calc class we just had to do the math, and i would multiply the exponent to the coefficient and minus the exponent by 1. I'm 99.9% this is correct.

    But my econ teacher wants us to caclucate to sets:
    a. calculate derivative Q/derivative L
    b. calculate derivative Q/derivative K

    I'm lost on that part. Thanks
     \frac{dQ}{dK} means you differentiate the equation with respect to variable K, and present that all other variables are constants!  \frac{dQ}{dL} means the same thing, except you differentiate with respect to L, and present that K is a constant!

    SO!

     \frac{dQ}{dK} = \frac{d}{dQ} 10L^{\frac{3}{4}} K^{\frac{1}{4}}

    =    10L^{\frac{3}{4}}\frac{d}{dQ} (K^{\frac{1}{4}})

     =    10L^{\frac{3}{4}} ( \frac{1}{4}K^{\frac{-3}{4}})

     =    \frac{10}{4}L^{\frac{3}{4}} K^{\frac{-3}{4}}

     =    \frac{5}{2}L^{\frac{3}{4}} K^{\frac{-3}{4}}

     \frac{dQ}{dL} = \frac{d}{dL} 10L^{\frac{3}{4}} K^{\frac{1}{4}}

    =  K^{\frac{1}{4}} \frac{d}{dQ}(10L^{\frac{3}{4}})

    =  K^{\frac{1}{4}} (\frac{3}{4}10L^{\frac{-1}{4}})

    =  K^{\frac{1}{4}} (\frac{30}{4}L^{\frac{-1}{4}})

    =  K^{\frac{1}{4}} (\frac{15}{2}L^{\frac{-1}{4}})

    =  \frac{15}{2}K^{\frac{1}{4}} L^{\frac{-1}{4}}
    Last edited by mr fantastic; January 22nd 2009 at 04:22 AM. Reason: Fixed all the latex (6 lines). You owe me, Mush lol!
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