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Math Help - show that f ' is even..

  1. #1
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    show that f ' is even..

    Suppose that f is an odd function which is differential for all x. Show that f' is even.

    I know this is like an easy question but I'm totally forgetting how to right a formal proof for these because I haven't done them in so long...but I know like let f=2x+1 and then f' would be 2 which would be even..but I feel like I have the show the deriv as 2k not just 2 but I'm forgetting how to work through it lol thanks!
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    f'(-x) = \lim {f(-x+h)-f(-x)\over h} = \lim {f(x) - f(x-h)\over h}=f'(x) where I used that f is odd in the second step.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tn11631 View Post
    Suppose that f is an odd function which is differential for all x. Show that f' is even.

    I know this is like an easy question but I'm totally forgetting how to right a formal proof for these because I haven't done them in so long...but I know like let f=2x+1 and then f' would be 2 which would be even..but I feel like I have the show the deriv as 2k not just 2 but I'm forgetting how to work through it lol thanks!
    Alternatively: f'(-x)+f'(x)=(-f(-x)+f(x))'=(2f(x))'=2f'(x)
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    Quote Originally Posted by Drexel28 View Post
    Alternatively: f'(-x)+f'(x)=(-f(-x)+f(x))'=(2f(x))'=2f'(x)
    oh man, I was thinking way to far back into math reasoning..Thanks guys!
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tn11631 View Post
    oh man, I was thinking way to far back into math reasoning..Thanks guys!
    By the way, I wrote that stupidly.

    f'(-x)=(-f(-x))'=(f(x))'<br />
=f'(x)
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    Now you lost a prime! I liked it the first way anyway ;p
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  7. #7
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by maddas View Post
    Now you lost a prime! I liked it the first way anyway ;p
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