I am learning about directional derivative and the definition bothered me so much.
$\frac{\partial f}{\partial \vec{v}} = \vec{\nabla}f . \vec{v}$
which by extension can be written as
$\frac{\partial f}{\partial \vec{v}} = (\vec{\nabla}f . \hat{v})||\vec{v}||$
So that would mean derivative of $f$ along $\vec{2x}$ will be
$\frac{\partial f}{\partial \vec{2x}} = (\vec{\nabla}f . \hat{i})2 = 2\frac{\partial f}{\partial x}$
but $\frac{\partial f}{\partial(2x)}$ should be $\frac{1}{2}\frac{\partial f}{\partial x}$
Why there is a difference?
Note: I am beginner to multivariate calculus, so the question is probably a silly misunderstanding of the concepts. I will be glad if you can clarify it. Thank you.
The way you should think about the directional derivative is this: Take any parametrized curve through $p$ with velocity vector $\vec v$ at $p$. Compute the rate of change of $f$ along this curve at the instant you pass through $p$. (Then it's clear that if you're moving twice as fast, the function is changing twice as fast at that instant.)
COMMENT: Personally, I prefer the notation $D_{\vec v}f(p)$ for the directional derivative.