# Thread: show that f ' is even..

1. ## 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!

2. $\displaystyle f'(-x) = \lim {f(-x+h)-f(-x)\over h} = \lim {f(x) - f(x-h)\over h}=f'(x)$ where I used that f is odd in the second step.

3. 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: $\displaystyle f'(-x)+f'(x)=(-f(-x)+f(x))'=(2f(x))'=2f'(x)$

4. Originally Posted by Drexel28
Alternatively: $\displaystyle f'(-x)+f'(x)=(-f(-x)+f(x))'=(2f(x))'=2f'(x)$
oh man, I was thinking way to far back into math reasoning..Thanks guys!

5. 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.

$\displaystyle f'(-x)=(-f(-x))'=(f(x))' =f'(x)$

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

7. Originally Posted by maddas
Now you lost a prime! I liked it the first way anyway ;p