I have a challenging question for homework. I have done implicit differentiation before in the textbook run-of-the-mill fashion but I do not know how one would go about setting up this an equation for a question like this:
Another user told me to set it up as $(DY/DX)(DX/DT) = (DY/DT)$ , but if that is correct, I do not understand how this was arrived at. Can anyone provide an example or explanation?!
Yes, by the chain rule, it is true that given a function $y(x(t))$ $$\dfrac{dy}{dx}\cdot \dfrac{dx}{dt} = \dfrac{dy}{dt}\tag{1}$$
It also (then) follows that $$\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}\tag{2}$$
So given functions $y(t), x(t)$ we can compute, separately, $\dfrac{dy}{dt}$, $\dfrac{dx}{dt}$, to obtain using $(2)$: $\dfrac{dy}{dx}$.
We can then find $\dfrac {d^2y}{dx^2}$.