# Thread: Odd/Even Function Derivative Quesiton

1. ## Odd/Even Function Derivative Quesiton

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 ?

2. 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 ?
$\displaystyle y = f(-x) \Rightarrow \frac{dy}{dx} = - f'(-x)$.

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

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

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

3. 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)

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

Thank you I understand now