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Question Number 95635 by Ar Brandon last updated on 26/May/20

$$\mathrm{Show}\mathrm{that}\mathrm{if}\mathrm{a}\mathrm{function}\mathrm{is}\mathrm{even}\mathrm{and}\mathrm{derivable}\mathrm{then}$$$${\mathrm{f}}^{\prime }\left(\mathrm{x}\right)\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{function}.$$

Answered by mr W last updated on 26/May/20

$$f\left(x\right)iseven:f(-x)=f\left(x\right)$$$$f^{\prime }(-x)=\frac{df(-x)}{d(-x)}=-\frac{df\left(x\right)}{dx}=-f^{\prime }\left(x\right)$$$$\Rightarrow f^{\prime }\left(x\right)isodd.$$

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