what is the formular of k-total derivative of vector-valued function

Question

I am reading the book "Mathematics for machine learning". On chapter 5.8 they talk about Taylor series like this: $\text{For a function}\\ f: R^{n} \to R \\x \to f(x) , x ∈ R^{n} \\\space \text{that f is smooth at } x_{0}. \space \space\text{When we define the difference vector} \\ \alpha:=x-x_{0} \space \text{, the multivariate Taylor series of f at } x_{0} \space \text{is defined as} \\f(x) = \sum_{k=0}^{\infty}\frac{D^{k}_{x}f(x_{0})}{k!}.\alpha^{k} \\\text{where }\space \space D^{k}_{x}f(x_{0}) \space \text{is the k-th (total) derivative of f with respect to x, evaluated at} \space x_{0}. \space \\ D^{k}_{x}f \space \text{and } \alpha^{k} \space \space \text{are k-th order tensors , i.e., k-dimensional arrays. The kth-order tensor} \space \alpha^{k} \in R^{n^{k}} \space \text{is obtained as a k-fold outer product, denoted by} \otimes \space \text{, for example} \space \alpha^{2}=\alpha\otimes\alpha, \space \alpha^{3}=\alpha\otimes\alpha\otimes\alpha,........\\ \text{But what is } D^{k}_{x}f ?$ $\\ \text{for example x}\in R^{2}; x=\left[ \begin{matrix} x_{1} \\ x_{2} \end{matrix} \right], \text{what is the formular of } D^{k}_{x}f \space ?$