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Homework Help: Partial derivatives Q involving homogeneity of degree n

  1. Mar 10, 2015 #1
    1. The problem statement, all variables and given/known data

    Show that if f is homogeneous of degree n, then

    [tex] x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf(x,y) [/tex]

    Hint: use the Chain Rule to diff. f(tx,ty) wrt t.

    2. The attempt at a solution

    I know that if f is homogeneous of degree n then [tex] t^nf(x,y) = f(tx,ty) [/tex]

    But I'm really at a conceptual loss here.

    I've tried the hint, I let f(tx, ty) = f(a,b) so a = tx and b = ty then

    [tex] \frac{\partial f}{\partial t} = \frac{\partial f}{\partial a}\frac{\partial a}{\partial t}+\frac{\partial f}{\partial b}\frac{\partial b}{\partial t} \\
    \frac{\partial f}{\partial t} = \frac{\partial f}{\partial a}x + \frac{\partial f}{\partial b}y [/tex]

    And I have tried looking at

    [tex] \frac{d}{dt}t^nf(x,y) = nt^{n-1}f(x,y) [/tex]

    But beyond this I just cannot see what I need to do.

  2. jcsd
  3. Mar 10, 2015 #2


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  4. Mar 10, 2015 #3


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    Gold Member

    That's a good start. I'll call the first equation in your attempt (1) and the last one (2).
    Now in (2) you are taking the derivative (with respect to t) of the left hand of (1). Do the same thing with the right hand and see where that leads you.
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