I'm learning how to take second order derivatives of parametric equations with respect to $x$, and I'm a little confused about the why of the process and what the formulas represent.
Assume we are talking about positional vectors changing with time. I understand that the following represents the position of $f(t)$ over time:
$$ f(t) = \left[x(t), y(t)\right] $$
It is also my understanding that the parametric derivative represents the velocity and magnitude (with respect to t) at a specific time:
$$ \frac{d}{dt} f(t) = \left[\frac{d}{dt}x(t), \frac{d}{dt}y(t)\right] $$
I believe that this can be represented more succinctly like this:
$$ \frac{df}{dt} = \left[\frac{dx}{dt}, \frac{dy}{dt}\right] $$
Now, if we want $\frac{dy}{dx}$ (aka instantaneous velocity with respect to x --or-- the slope of the tangent of the curve with no regard to magnitude) then we simply divide the y portion by the x portion, like this:
$$ \frac {\left[\frac{dy}{dt}\right]}{\left[\frac{dx}{dt}\right]} = \left[\frac{dy}{dx}\right] $$
This is all first order, and I believe I understand it. Now we get to second order, and I can't quite wrap my head around it. I've been told that the second order derivative -- instantaneous acceleration with respect to $x$ -- is:
$$ \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left[\frac{dy}{dx}\right]}{\left[\frac{dx}{dt}\right]} $$
I can see that the units sort of work out if you pretend that these are fractions. And I suppose I understand that you need to divide by another $ \frac{dx}{dt} $ in order to remove the effect of $t$ and get only the change in $x$.
However, I'm confused about why other methods don't work, and what other functions actually represent. Firstly, what is it I'm getting when I just take the derivative of $\frac{dy}{dx}$? Why isn't it the change in slope with respect to x? And how would that be representing in this notation? Does the following represent any coherent concept?
$$ \frac{d}{dt} \left[\frac{dy}{dx}\right] = \frac{d^2y}{dxdt} ?? $$
More confusingly, does this represent the parametric acceleration and magnitude at time t:
$$ \frac{d^2f}{dt^2} = \left[\frac{d^2x}{dt^2}, \frac{d^2y}{dt^2}\right] $$
And if so, what does this represent and why isn't it the same as $ \frac{d^2y}{dx^2} $:
$$ \frac{\left[\frac{d^2y}{dt^2}\right]}{\left[\frac{d^2x}{dt^2}\right]} = \left[\frac{d^2y}{d^2x}\right] ?? $$
What does $ \left[\frac{d^2y}{d^2x}\right] $ represent? Isn't it just the slope of the acceleration vector? How is that not the 2nd derivative of $ y $ with respect to $ x $? I understand that it isn't, I'm just confused as to why not. What is the verbal meaning of these things?