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Differentiation Proof

  1. Sep 24, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove if f(x) is differentiable at x=a then [tex]f'(a)=lim_{h\rightarrow0}\frac{f(a+h)-f(a-h)}{2h}[/tex]


    2. Relevant equations
    I know that the derivative is defined as
    [tex]f'(a)=lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}[/tex]


    3. The attempt at a solution
    Starting from the definition I used a known relation.
    [tex]f'(a)=lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}=lim_{h\rightarrow0}\frac{f(a+h)-f(a)}{h}
    =lim_{h\rightarrow0}\left[\frac{f(a+h)-f(a-h)}{h}+\frac{f(a-h)-f(a)}{h}\right][/tex]

    I'm not sure how to decompose anymore from here. Any help you could give would be greatly appreciated.
     
  2. jcsd
  3. Sep 24, 2009 #2

    tiny-tim

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    Science Advisor
    Homework Helper

    Hi dgonnella89! :smile:

    Hint: put k = -h in the definition of f'(a). :wink:
     
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