I am having trouble with a term that arises in a physics equation (deriving the Lagrange equation for one particle in one generalized coordinate, $q$, dimension from one Cartesian direction, $x$).
My struggle essentially boils down to manipulating the following expression in one of the terms, $$\frac{d}{dt}\left[\frac{dx(q)}{dq}\right],$$ where $x$ is a function of only $q$. I know the order of partial derivatives is interchangeable; is it also true here for total derivatives?
if $x=x(q$ only$)$ then \begin{equation}\frac{dx}{dq}=\frac{\partial x}{\partial q} \end{equation} therefore, \begin{equation}\frac{d}{dt}\frac{dx}{dq}=\frac{d}{dt}\frac{\partial x}{\partial q}= \frac{\partial \dot{x}}{\partial q}=\frac{\partial}{\partial q} \frac{dx}{dt}\neq\frac{d}{dq}\frac{dx}{dt} \end{equation}
Inequality in the end is because $\frac{dx}{dt}$ can be an explicit function of both $q$ and $\dot{q}$, in which case we cannot swap total derivative w.r.t $q$ with a partial one.
Be careful, even though $x$ is a function of $q$ only, $\frac{dx}{dt}$ will still not be zero as it's a total derivative and $x$ may have some implicit time dependence through $q$. That being said, $\frac{\partial x}{\partial t}$ is equal to zero as $x$ does not depend on time explicitly.
