show that f ' is even..

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April 13th 2010, 06:10 PM
tn11631

show that f ' is even..

Suppose that f is an odd function which is differential for all x. Show that f' is even.

I know this is like an easy question but I'm totally forgetting how to right a formal proof for these because I haven't done them in so long...but I know like let f=2x+1 and then f' would be 2 which would be even..but I feel like I have the show the deriv as 2k not just 2 but I'm forgetting how to work through it lol thanks!
April 13th 2010, 06:39 PM
maddas

$f'(-x) = \lim {f(-x+h)-f(-x)\over h} = \lim {f(x) - f(x-h)\over h}=f'(x)$ where I used that f is odd in the second step.
April 13th 2010, 06:45 PM
Drexel28

Quote:

Originally Posted by tn11631

Suppose that f is an odd function which is differential for all x. Show that f' is even.

I know this is like an easy question but I'm totally forgetting how to right a formal proof for these because I haven't done them in so long...but I know like let f=2x+1 and then f' would be 2 which would be even..but I feel like I have the show the deriv as 2k not just 2 but I'm forgetting how to work through it lol thanks!

Alternatively: $f'(-x)+f'(x)=(-f(-x)+f(x))'=(2f(x))'=2f'(x)$
April 13th 2010, 06:57 PM
tn11631

Quote:

Originally Posted by Drexel28

Alternatively: $f'(-x)+f'(x)=(-f(-x)+f(x))'=(2f(x))'=2f'(x)$

oh man, I was thinking way to far back into math reasoning..Thanks guys!
April 13th 2010, 06:59 PM
Drexel28

Quote:

Originally Posted by tn11631

oh man, I was thinking way to far back into math reasoning..Thanks guys!

By the way, I wrote that stupidly.

$f'(-x)=(-f(-x))'=(f(x))'<br /> =f'(x)$
April 13th 2010, 07:04 PM
maddas

Now you lost a prime! I liked it the first way anyway ;p
April 13th 2010, 07:06 PM
Drexel28

Quote:

Originally Posted by maddas

Now you lost a prime! I liked it the first way anyway ;p

(Swear)