Is there any book/article that gives a general result of this:
For any $n\in \mathbb{N}^*$, use the chain rule to compute the $n$-th order derivative of :$(f_1(x_1(t),x_2(t),\dots,x_n(t)),f_2(x_1(t),x_2(t)),\dots,x_n(t)),\dots,f_n(x_1(t),x_2(t),\dots,x_n(t)))$ respect to $t$. The results should be represented by partial derivatives of $\frac{d^k f_i}{x_j^k}$ and $\frac{d^k x_j(t)}{t^k} $ for any $k\in \mathbb{N}^*$. Using tensor to do this might be a choice.
I have computed the 1st/2nd/3rd derivative in the form of tensor. However, i can not make sure that my result is 100% correct and the form of formula may not be neat enough.