Let $X$ be a smooth algebraic variety over a field $k$, and let $x\in X$. Let $(x_1,\dots,x_n)$ be a regular system of parameters at $x$, so that $\Omega^1_{X/k}$ is locally free at $x$ with basis $dx_1,\dots,dx_n$.
Whenever $f$ is a regular function defined on a (Zariski-)neighbourhood of $x$, one may define the partial derivatives $\frac{\partial f}{\partial x_i}(x)\in\kappa(x)$ of $f$ at $x$ as being the unique scalars of $\kappa(x)$ verifying $df(x)=\frac{\partial f}{\partial x_1}(x)dx_1+\dots+\frac{\partial f}{\partial x_n}(x)dx_n$ (where $df(x)$ is the image of $df$ in $(\Omega^1_{X/k})_x\otimes_{\mathscr{O}_{X,x}}\kappa(x)$).
Now, I would like to define higher partial derivatives of $f$ with respect to the $x_i$'s. How may this be done?