I am having some problems understanding directional derivatives and limit rules. I was hoping someone might be able to help.
Basically, the result is the $f'(x, -a)$ = $-f'(x,a)$. To show this (after some manipulations, we have:
(1) $f'(x, -a) = \lim\limits_{(-h) \to 0} \frac{f(x+(-h)a) - f(x)}{(-h)} = -f'(x,a)$ (2)
I had two questions:
Firstly, can I intepret the result as (1) the instantenous rate of change in the direction opposite to $a$ is equal to (2) the instantaneous rate of change in the opposite direction to $a$?
I don't understand the second equality. When I write out $-f(x,a)$ according to the definition, I have:
$-f'(x, a) = \lim\limits_{h \to 0} \frac{-f(x+ ha) + f(x)}{h}$
It's not obvious to me why the two are equal. Is it because $\lim\limits_{h \to 0}$ is equivalent to $\lim\limits_{(-h) \to 0}$?
Thank you.
The equality that you wish to prove means that the instantaneous rate of change in the opposite direction of $a$ is the symetric of the instantaneous rate of change in the direction of $a$. And the answer to your final question is affirmative.