even and odd

**flower3** · January 4th 2010, 11:14 PM

Recall that $f: \mathbb{R} \to \mathbb{R}$ is an even function if $f(-x)=f(x) , \ \forall x \in \mathbb{R}$
and is an odd function if $f(-x)=-f(x) , \ \forall x \in \mathbb{R}$
prove that if $f: \mathbb{R} \to \mathbb{R}$ is an even differentiable function on $\mathbb{R} \$
then $f'$ is an odd function

**tonio** · January 4th 2010, 11:22 PM

Originally Posted by flower3

Recall that $f: \mathbb{R} \to \mathbb{R}$ is an even function if $f(-x)=f(x) , \ \forall x \in \mathbb{R}$
and is an odd function if $f(-x)=-f(x) , \ \forall x \in \mathbb{R}$
prove that if $f: \mathbb{R} \to \mathbb{R}$ is an even differentiable function on $\mathbb{R} \$
then $f'$ is an odd function

$f'(x_0):=\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$ ...and the quotient of even by odd functions is an odd function.

Tonio

**HallsofIvy** · January 5th 2010, 04:04 AM

Use the chain rule: the derivative of f(-x) is f'(-x)(-x)'= -f'(-x). The derivative of -f(x) is -f'(x). Put those together.

**HallsofIvy** · January 5th 2010, 04:07 AM

Originally Posted by tonio

$f'(x_0):=\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$ ...and the quotient of even by odd functions is an odd function.

Tonio

I don't understand your point, tonio. If $g(x)= x- x_0$ , then $g(-x)= -x-x_0$ is not equal to either $g(x)= x-x_0$ nor $-g(x)= -x+ x_0$ so g is neither even nor odd. What even and odd functions make up your quotient?

**tonio** · January 5th 2010, 05:45 AM

Originally Posted by HallsofIvy

I don't understand your point, tonio. If $g(x)= x- x_0$ , then $g(-x)= -x-x_0$ is not equal to either $g(x)= x-x_0$ nor $-g(x)= -x+ x_0$ so g is neither even nor odd. What even and odd functions make up your quotient?

I wrote that too quick: I meant to write the definition of the derivative as

$f'(x_0)=\lim_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h}$ , and then

$f'(-x_0)=\lim_{h\to 0}\frac{f(-x_0+h)-f(-x_0)}{h}$ . Now, here substitute k:=-h, and of course $h\rightarrow 0 \Longleftrightarrow k\rightarrow 0$ , so then (and here enters the numerator even and the denominator odd thing)

$f'(-x_0)=\lim_{h\to 0}\frac{f(-x_0+h)-f(-x_0)}{h}=\lim_{k\to 0}\frac{f(-x_0-k)-f(-x_0)}{-k}=$ $\lim_{k\to 0}\frac{f(x_0+k)-f(x_0)}{-k}=-\lim_{k\to 0}\frac{f(x_0+k)-f(x_0)}{k}=-f'(x_0)$

Tonio

**Drexel28** · January 5th 2010, 06:58 AM

Originally Posted by flower3

Recall that $f: \mathbb{R} \to \mathbb{R}$ is an even function if $f(-x)=f(x) , \ \forall x \in \mathbb{R}$
and is an odd function if $f(-x)=-f(x) , \ \forall x \in \mathbb{R}$
prove that if $f: \mathbb{R} \to \mathbb{R}$ is an even differentiable function on $\mathbb{R} \$
then $f'$ is an odd function

Essentially the same thing as HallsOfIvy is to define $h(x)=f(x)-f(-x)=0$ and then note that $h'(x)=f'(x)+f'(-x)=0\implies f'(x)=-f'(-x)$

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