Odd/Even Function Derivative Quesiton

**StarlitxSunshine** · December 17th 2009, 03:51 PM

Suppose that g is a function with the following two properties g(-x) = g(x) for all x, and g'(a) exists. Which of the following must necessarily be equal to g'(-a) ?

I thought it would g'(-a), but the correct answer is -g'(a).

Why is this so, please ?

**mr fantastic** · December 17th 2009, 03:57 PM

Originally Posted by StarlitxSunshine

Suppose that g is a function with the following two properties g(-x) = g(x) for all x, and g'(a) exists. Which of the following must necessarily be equal to g'(-a) ?

I thought it would g'(-a), but the correct answer is -g'(a).

Why is this so, please ?

$y = f(-x) \Rightarrow \frac{dy}{dx} = - f'(-x)$ .

$y = f(x) \Rightarrow \frac{dy}{dx} = f'(x)$ .

Therefore $- f'(-x) = f'(x) \Rightarrow f'(-x) = - f'(x)$ .

If $f(x)$ is even then $f'(x)$ is odd.

**skeeter** · December 17th 2009, 03:58 PM

Originally Posted by StarlitxSunshine

Suppose that g is a function with the following two properties g(-x) = g(x) for all x, and g'(a) exists. Which of the following must necessarily be equal to g'(-a) ?

I thought it would g'(-a), but the correct answer is -g'(a).

Why is this so, please ?

g(-x) = g(x)

taking the derivative ...

-g'(-x) = g'(x)

multiply both sides by -1 ...

g'(-x) = -g'(x)

so ...

g'(-a) = -g'(a)

**StarlitxSunshine** · December 17th 2009, 03:59 PM

Oh :O
I think I got the sign mixed up while I was doing it ~

Thank you

I understand now

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