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Partial Derivatives With N-Variables

  1. Oct 25, 2011 #1
    1. The problem statement, all variables and given/known data

    Given F(x_1,x_2,...,x_i,...,x_n) = nth-root(x_1*x_2*...*x_i*...*x_n), how do I take the partial derivative with respect to x_i, where x_i is an arbitrary variable?

    2. Relevant equations



    3. The attempt at a solution

    Would it just be:

    (1/n)(x_1*x_2*...*x_i*...*x_n)^((1/n)-1)*(x_1*x_2*...*x_i-1*x_i+1*...*x_n)?
     
  2. jcsd
  3. Oct 25, 2011 #2
    You got it!
     
  4. Oct 25, 2011 #3
    Alright, nice!

    What I'd like to try to do with this problem is maximize F (the equation for the geometric mean) when it is constrained by G(x_1,x_2,...,x_n) = x_1 + x_2 + ... + x_n = c, where c is some constant. I can take the partial derivative of G with respect to x_i and get,

    G_x_i = 1

    But I don't really know if I am a) on the right track for this problem or b) how to proceed if I am on the right track. What should my next step be?
     
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