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Partial derivative

  1. Mar 18, 2009 #1
    1. The problem statement, all variables and given/known data
    A function f: R^n--R is homogenous of degree p if f( [tex]\lambda[/tex]x)=[tex]\lambda[/tex]^p f(x) for all [tex]\lambda[/tex][tex]\in[/tex]R and all x[tex]\in[/tex]R^n
    show that if f is differentiable at x ,then x[tex]\nabla[/tex]f(x)=pf(x)



    2. Relevant equations



    3. The attempt at a solution
    set g([tex]\lambda[/tex])=f([tex]\lambda[/tex]x)
    find out g'(1)
    then how to continue ?
     
  2. jcsd
  3. Mar 19, 2009 #2
    any help?
     
  4. Mar 19, 2009 #3

    Office_Shredder

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    [tex] g( \lambda _ = f( \lambda x)[/tex]

    Then [tex]g'( \lambda ) = \sum_{i=1}^{n} \frac{df}{dx_i} \frac{d( \lambda x}{dx_i}[/tex]

    The right hand side is obtained using the chain rule. Try to calculate what the right hand side really is
     
  5. Mar 19, 2009 #4
    i don't know where is the formula for g' comes from
     
  6. Mar 20, 2009 #5

    HallsofIvy

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    Office Shredder told you: it is the chain rule.
     
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