Comments on: Derivatives of Even Functions

By: MrHonner

MrHonner — Thu, 20 Sep 2012 01:58:45 +0000

Yes, that is probably the quickest proof. I shared the original proof because I really like that trick of splitting up every function an even part and an odd part (here, since the function here is even to begin with, it has only an even part).

I don’t think either of these algebraic proofs are very illuminating, though, which is why I like the tangent line argument.

By: Nat

Nat — Wed, 19 Sep 2012 22:00:36 +0000

hmm..
What about writing f(x) = f(-x) (by definition)?
Differentiating both sides gives us
f’(x) = f’(-x)*(-1) (by the Chain Rule).
So by definition f’ is odd.

By: MrHonner

MrHonner — Fri, 14 Sep 2012 21:00:23 +0000

I think a more natural proof would show that the tangents to an even function at x = c and x = -c have opposite slopes. This probably follows pretty quickly from the definition of derivative.

By: Oliver Prior

Oliver Prior — Fri, 14 Sep 2012 20:46:37 +0000

Impressive derivation, I enjoyed it. Any other similar proofs you’d recommend?

By: MrHonner

MrHonner — Fri, 14 Sep 2012 20:05:40 +0000

Sue-

I factored out a (-1) from the numerator. Now rearrange the terms in the numerator and you’ll have (f’(x) – f’(-x)), which is the numerator of a’(x).

Sorry if I sacrificed a bit of clarity for brevity!

By: Sue VanHattum

Sue VanHattum — Fri, 14 Sep 2012 18:31:29 +0000

Last line, after 3rd = sign, I’m not seeing it.