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Derivative of summation

  1. Apr 10, 2010 #1
    Hello,

    Could you help me derive this function, so I can find the minimum of it.

    [tex]z=\sum_{i=1}^{n}{\sqrt{\left( x-x_{i} \right)^{2}+\left( y-y_{i} \right)^{2}}}[/tex]

    Thank you.
     
  2. jcsd
  3. Apr 10, 2010 #2

    D H

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    This looks like homework. We help you do your own homework; we do not do it for you.

    You need to show some work before someone will help you.
     
  4. Apr 10, 2010 #3
    It is not homework. It is just some curiosity of mine.

    What I want to do is find the point [tex](x,y)[/tex], that has the smallest sum of distances to a series of points [tex](x_{1},y_{1})[/tex], [tex](x_{2},y_{2})[/tex], [tex](x_{3},y_{3})[/tex], ...,[tex](x_{n},y_{n})[/tex]. Something like a centre of gravity.

    I don't need just the result, I would like to see the path to it.

    Thank you.
     
  5. Apr 10, 2010 #4

    Redbelly98

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    Welcome to Physics Forums.
    Thanks for the clarification, it makes a difference in how we approach helping you. There are designated homework subforums (not this one however) that some new members ignore.

    First, realize that there is not necessarily a unique solution to this. Consider a set of just 2 points. Any point on the line segment joining them will have the same sum-of-distances.

    That being said, you would take the partial derivatives of z with respect to both x and y, set each equal to zero, and solve the two equations you get.
     
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