As far as I know, there are two solutions to this problem. The first solution is to transform the equation as $$\dfrac{dx}{d\dot{x}} = \dfrac{dx}{dt}\dfrac{dt}{d\dot{x}} = \dfrac{\dot{x}}{\ddot{x}}$$ Yet when I was viewing lecture 15 of the engineering dynamics lectures of MIT here, when talking about the Lagrange method, I saw the lecturer simply evaluate this derivative to 0 because of the displacement, x is apparently not a function of speed, $\dot{x}$. (it may have been in regards to the angle, $\theta$)
I have followed the second approach for the most part yet I was required to Taylor expand a certain differential equation(which contains x) with respect to $\dot{x}$ while doing a physics equation. In this case, should I evaluate $\dfrac{dx}{d\dot{x}}$ to 0 or $\dfrac{\dot{x}}{\ddot{x}}$? If the latter is the case, was I ever correct in evaluating $\dfrac{dx}{d\dot{x}}$ to 0?
As a reference, $\ddot{x}$ is a function of $x$ and $\dot{x}$ in my differential equation.
As stated in the comments, it is a partial derivative $$\frac{\partial x}{\partial\dot x}$$If $x$ does not have any explicit dependence on $\dot x$, then this is $0$, which is what happens in this case. If it was a total derivative $\frac{dx}{d\dot x}$ where $x$ was just a function of $\dot x$, then yes, it would be $$\frac{dx}{d\dot x}=\frac{\dot x}{\ddot x}$$