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Math Help - jacobian matrix. derivatives

  1. #1
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    Question jacobian matrix. derivatives

    calculate the jacobian matrix of the function at the given point. and write the formula for the total derivative.
    M(x1,x2,x3,x4)= ( (x1+x2)/(x3+x4), (x3+x4)/(x1+x2) )
    a= (1,1,1,1)

    how do i set up the matrix for this? any hints would be greatly appreciated to start me off
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  2. #2
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    edit

    okay so i set it up as a 2x4 matrix:
    dM1/dx1 dM1/dx2 dM1/dx3 dM1/dx4
    dM2/dx1 dM2/dx2 dM2/dx3 dM2/dx4

    but from there i got confused. for example in the first one, how do i take the partial derivative of (x1+x2)/(x3+x4) with respect to x1? wouldn't it be like 1/0 ? then where does the point (1,1,1,1) come into play? ah i'm so confused..
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by holly123 View Post
    okay so i set it up as a 2x4 matrix:
    dM1/dx1 dM1/dx2 dM1/dx3 dM1/dx4
    dM2/dx1 dM2/dx2 dM2/dx3 dM2/dx4

    but from there i got confused. for example in the first one, how do i take the partial derivative of (x1+x2)/(x3+x4) with respect to x1? wouldn't it be like 1/0 ? then where does the point (1,1,1,1) come into play? ah i'm so confused..
    Think of x_2,x_3,x_4 as constants when differentiating with respect to x_1. So \frac{\partial}{\partial x_1}\left[\frac{x_1+x_2}{x_3+x_4}\right]\sim \frac{\,d}{\,dx_1}\left[\frac{x_1}{b}+\frac{a}{b}\right]=\frac{1}{b}. So \frac{\partial}{\partial x_1}\frac{x_1+x_2}{x_3+x_4}=\frac{1}{x_3+x_4}.

    Can you use this same idea to differentiate the others?
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  4. #4
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    hmm okay so it would be

    (1/x3+x4) (1/x3+x4) (x1+x2) (x1+x2)
    (x3+x4) (x3+x4) (1/x1+x2) (1/x1+x2)

    i think?

    and you plug in (1,1,1,1) to get
    1/2 1/2 2 2
    2 2 1/2 1/2


    is that right?
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