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Proof with partial derivatives

  1. Oct 11, 2009 #1
    I've been trying to prove that if the following statement holds for all (x,y)ER^2, f must be a linear function:

    f(x,y)-f(0,0)=x*(d/dx)[f(x,y)]+y*(d/dy)[f(x,y)]

    It seems to work for any function I plug in, but I'm unable to establish why this always works. Also, when I say (d/dx)[f(x,y)], I mean the derivative of f(x,y) with respect to x.

    Thanks in advance for any help or ideas.
     
  2. jcsd
  3. Oct 12, 2009 #2

    arildno

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    IF f(x,y) is a LINEAR function, then you know thay f(x,y)=Ax+By+C, with A, B and C being constants.

    Thus, f(x,y)-f(0,0)=Ax+By.

    Does that help?
     
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