# derivative of even/odd function (proof)

Suppose $f(x)=\pm f(-x)$. We need to show that
${f}^{\prime}(x)=\mp {f}^{\prime}(-x)$.
To do this, let us define the auxiliary function
$m:\mathbb{R}\to \mathbb{R}$, $m(x)=-x$.
The condition on $f$ is then $f(x)=\pm (f\circ m)(x)$.
Using the chain rule^{}, we have that

${f}^{\prime}(x)$ | $=$ | $\pm {(f\circ m)}^{\prime}(x)$ | ||

$=$ | $\pm {f}^{\prime}\left(m(x)\right){m}^{\prime}(x)$ | |||

$=$ | $\mp {f}^{\prime}(-x),$ |

and the claim follows. $\mathrm{\square}$

Title | derivative of even/odd function^{} (proof) |
---|---|

Canonical name | DerivativeOfEvenoddFunctionproof |

Date of creation | 2013-03-22 13:37:57 |

Last modified on | 2013-03-22 13:37:57 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Proof |

Classification | msc 26A06 |