Second derivative of parametric equations

Question

I am looking for an intuitive explanation for the formula used to take the second derivative of a parametric function. The formula is:

$\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$

I understand the reasoning for getting $\frac{dy}{dx}$-- by dividing $\frac{dy}{dt}$ by $\frac{dx}{dt}$ -- however I am lost in the above formula. If we are finding the second derivative of y with respect to x, why do we differentiate $\frac{dy}{dx}$ with respect to t, but not $\frac{dx}{dt}$?

My inability to understand this could be due to my already fragile understanding of Leibnitz notation. At any rate, I would appreciate an explanation for why this formula is set up in this way.

Again, the $\frac{dy}{dx}$ part makes sense, we were finding the derivative of y with respect to x, and since both x and y are functions of t, we found how x changes with respect to t, and how y changes with respect to t, and divided them.

Thanks in advance for any help!

Answer 1

Intuitively (from Leibnitz notation): $$ \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}=\frac{d}{dt}\left(\frac{dy}{dx} \right)\frac{dt}{dx} = \frac{d}{dx}\left(\frac{dy}{dx} \right) $$

Answer 2

I asked my teacher and figured it out: since we want to find the derivative of $\frac{dy}{dx}$, we must find the derivative with respect to x, not t. Even though $\frac{dy}{dx}$ is in terms of t, when we find the derivative it must be with respect to x. This should make sense. If you wanted to find the derivative of 15x, you would apply the $\frac{d}{dx}$ operation, not something like $\frac{d}{dt}$ or $\frac{d}{dh}$ or any other variable. Since we now know we must use the $\frac{d}{dx}$ on $\frac{dy}{dx}$, it becomes clear that the $\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ is the same as using $\frac{d}{dx}$, since the dt's cancel.