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Question
Given a function $L(t, x_1(t), \cdots, x_n(t))$, its total derivative with respect to $t$ is given by
\[\frac{dL}{dt} = \frac{\partial L}{\partial t} + \sum_{j=1}^{n} \frac{\partial L}{\partial x_j} \frac{dx_j}{dt}\]which is a consequence of the chain rule. Note that I have dropped the function notation reminding us that we’re considering a collection of functions of $t$.
Is there a similar-ish expansion for the $k$th order derivative of $L(t, x_1(t), \cdots, x_n(t))$, denoted $\frac{d^k L}{dt^k}$?
For starters, we could look at the second application of the total derivative:
\[\begin{align} \frac{d}{dt} \frac{dL}{dt} = \frac{d^2L}{dt^2} &= \frac{d}{dt} \left[ \frac{\partial L}{\partial t} + \sum_{j=1}^{n} \frac{\partial L}{\partial x_j} \frac{dx_j}{dt} \right] \\ &= \frac{d}{dt} \left[ \frac{\partial L}{\partial t} \right] + \frac{d}{dt} \left[ \sum_{j=1}^{n} \frac{\partial L}{\partial x_j} \frac{dx_j}{dt} \right] \\ &= \frac{d}{dt} \left[ \frac{\partial L}{\partial t} \right] + \sum_{j=1}^{n} \frac{d}{dt} \left[\frac{\partial L}{\partial x_j} \frac{dx_j}{dt} \right] \\ &= \frac{d}{dt} \left[ \frac{\partial L}{\partial t} \right] + \sum_{j=1}^{n} \left(\frac{d}{dt} \left[\frac{\partial L}{\partial x_j} \right] \frac{dx_j}{dt} + \frac{\partial L}{\partial x_j} \frac{d}{dt} \left[\frac{dx_j}{dt} \right] \right) \\ \end{align}\]but I have not seen a way to simplify the above and generalize to finite $k$.