Results 1 to 5 of 5

Math Help - odd function prove

  1. #1
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    odd function prove

    if f(x) is an even function
    prove that f'(x) is odd function
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    12
    It's quite simple since f(x)=-f(-x), now just differentiate.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member ursa's Avatar
    Joined
    Jan 2009
    From
    Delhi
    Posts
    60
    It's quite simple since now just differentiate.
    krizalid you make a mistake
    its an even function
    so
    f(-x)=f(x), now differentiate
    -f'(-x)=f'(x)
    f'(-x)=-f'(x)
    hence, f'(x) is an odd function
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    12
    I let myself go with the word "odd," haha; anyway, the idea is the same.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Alternatively

    f'(x)=\lim_{\phi \to x}\frac{f(\phi)-f(x)}{\phi-x}

    So

    \begin{aligned}f'(-x)&=\lim_{\phi\to -x}\frac{f(\phi)-f(-x)}{\phi+x}\\<br />
&=\lim_{\phi\to -x}\frac{f(\phi)-f(x)}{\phi+x}\\<br />
&=\lim_{\phi\to x}\frac{f(-\phi)-f(x)}{-\phi+x}~~{\color{red}\star}\\<br />
&=-\lim_{\phi\to x}\frac{f(\phi)-f(x)}{\phi-x}\\<br />
&=-f'(x)\quad\blacksquare\end{aligned}

    {\color{red}\star}:\phi\mapsto -\phi
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to prove x^2 is a function?
    Posted in the Pre-Calculus Forum
    Replies: 12
    Last Post: October 8th 2011, 03:18 AM
  2. Prove a function is onto
    Posted in the Discrete Math Forum
    Replies: 29
    Last Post: April 30th 2010, 11:46 AM
  3. Prove a function is Onto
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: April 7th 2010, 08:01 AM
  4. Prove that f(x) is an odd function?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 21st 2009, 02:09 PM
  5. Prove a function is onto
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: April 12th 2009, 02:44 PM

Search Tags


/mathhelpforum @mathhelpforum