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If for a function f the equation f(x) = f( -x ) is true, is it also automatically true for all its derivatives and antiderivatives?

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    No. e.g. $f(x)=\cos x$.2012-04-24
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    Just derive and obtain $f'(x) = - f'(-x)$, so $f'$ is origin-symmetric as are derivatives of odd order, where derivatives of even order are axis-symmetric.2012-04-24

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