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Prove that f'(0) = 0

  1. Jun 16, 2011 #1
    1. The problem statement, all variables and given/known data
    If f(x) is an even function and f'(x) exists for all x, prove that f'(0) = 0. (Hint: Start with an equation that is true for all even functions and differentiate both sides with respect to x.)


    2. Relevant equations
    Equation true for all even functions: f(x) = f(-x)


    3. The attempt at a solution

    f(x) = f(-x)

    f(d/dx (x)) = f(d/dx (-x))

    f(1) = f(-1)

    I'm not sure if I have the notation correct when differentiating...or I could have done something like this:

    f(x) = f(-x)

    f' (x) = f' (-x)

    f'(0) = f'(-0)

    Then I'm not too sure as to what to do from there ):

    Thanks for all the help <3
     
  2. jcsd
  3. Jun 16, 2011 #2

    Dick

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    The derivative of f(x) is f'(x), sure. The derivative of f(-x) isn't equal to f'(-x). You need to use the chain rule to differentiate f(-x).
     
  4. Jun 16, 2011 #3
    But I'm taking the derivative with respect to x.... ): why do I need to use the chain rule?
     
  5. Jun 16, 2011 #4

    Dick

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    Because f(-x) has the form f(g(x)) where g(x)=(-x). Doesn't it?
     
  6. Jun 16, 2011 #5
    IF I am doing this right....

    f(x) = f(-x)

    f' (x) = f'(-x) + f(-1) derivative of outer function times inner function plus outside function times derivative of inside function o_O?

    Thank you for your patience (:!
     
  7. Jun 16, 2011 #6

    eumyang

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    No, I think you're confusing with the product rule.
     
  8. Jun 16, 2011 #7

    Dick

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    d/dx of f(g(x)) is f'(g(x))*g'(x), right? Isn't that the chain rule? If you put g(x)=(-x) what do you get for the derivative of f(-x)??
     
  9. Jun 16, 2011 #8
    Goodness I hope I'm doing it right this time, thanks for bearing with me.

    f'(x) = f'(g(x)) * g'(x)

    f'(x) = f'(-x) * (-1)

    f'(x) = -f'(-x).....= f'(x)?

    God I'm feeling so dumb right now ):!!!!
     
  10. Jun 16, 2011 #9

    Dick

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    You shouldn't feel dumb now that you are getting it right. Sure, f'(x)=(-f'(-x)). Put x=0 and tell me what f'(0) must be.
     
  11. Jun 16, 2011 #10
    God you're good <3

    Thank you so much (:!!!!
     
  12. Jun 17, 2011 #11

    HallsofIvy

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    Of course, he is!
     
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