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[parent] derivative of even/odd function (proof) (Proof)

Suppose $ f(x)=\pm f(-x)$. We need to show that $ f'(x)=\mp f'(-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

$\displaystyle f'(x)$ $\displaystyle =$ $\displaystyle \pm(f\circ m)'(x)$  
  $\displaystyle =$ $\displaystyle \pm f'\big(m(x)\big) m'(x)$  
  $\displaystyle =$ $\displaystyle \mp f'(-x),$  

and the claim follows. $ \Box$



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Cross-references: chain rule, function
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This is version 2 of derivative of even/odd function (proof), born on 2003-05-13, modified 2003-05-13.
Object id is 4281, canonical name is DerivativeOfEvenoddFunctionProof.
Accessed 8163 times total.

Classification:
AMS MSC: 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
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