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Math Help - Odd/Even Function Derivative Quesiton

  1. #1
    Junior Member StarlitxSunshine's Avatar
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    Odd/Even Function Derivative Quesiton

    Suppose that g is a function with the following two properties g(-x) = g(x) for all x, and g'(a) exists. Which of the following must necessarily be equal to g'(-a) ?

    I thought it would g'(-a), but the correct answer is -g'(a).

    Why is this so, please ?
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  2. #2
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    Quote Originally Posted by StarlitxSunshine View Post
    Suppose that g is a function with the following two properties g(-x) = g(x) for all x, and g'(a) exists. Which of the following must necessarily be equal to g'(-a) ?

    I thought it would g'(-a), but the correct answer is -g'(a).

    Why is this so, please ?
    y = f(-x) \Rightarrow \frac{dy}{dx} = - f'(-x).

    y = f(x) \Rightarrow \frac{dy}{dx} = f'(x).

    Therefore - f'(-x) = f'(x) \Rightarrow f'(-x) = - f'(x).

    If f(x) is even then f'(x) is odd.
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  3. #3
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    Quote Originally Posted by StarlitxSunshine View Post
    Suppose that g is a function with the following two properties g(-x) = g(x) for all x, and g'(a) exists. Which of the following must necessarily be equal to g'(-a) ?

    I thought it would g'(-a), but the correct answer is -g'(a).

    Why is this so, please ?
    g(-x) = g(x)

    taking the derivative ...

    -g'(-x) = g'(x)

    multiply both sides by -1 ...

    g'(-x) = -g'(x)

    so ...

    g'(-a) = -g'(a)
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  4. #4
    Junior Member StarlitxSunshine's Avatar
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    Oh :O
    I think I got the sign mixed up while I was doing it ~

    Thank you I understand now
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