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

Show that if a function is even and derivable then  f′(x) is an odd function.

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

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

f(x) is even: f(−x)=f(x)  f′(−x)=((df(−x))/(d(−x)))=−((df(x))/dx)=−f′(x)  ⇒f′(x) is odd.

$${f}\left({x}\right)\:{is}\:{even}:\:{f}\left(−{x}\right)={f}\left({x}\right) \\ $$$${f}'\left(−{x}\right)=\frac{{df}\left(−{x}\right)}{{d}\left(−{x}\right)}=−\frac{{df}\left({x}\right)}{{dx}}=−{f}'\left({x}\right) \\ $$$$\Rightarrow{f}'\left({x}\right)\:{is}\:{odd}. \\ $$

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