Confused on Lie Derivative of a Lie Derivative

Question

For my analysis class, we are using a differential forms textbook and we have defined the Lie derivative. There are two versions that my book uses, which is $L_{\textit{v}}f(p) = Df(p)v$ and $L_{\textit{v}}f = \displaystyle \sum_{i=1}^n g_i\frac{\partial f}{\partial x_i}$ where $\textit{v} = \displaystyle \sum_{i=1}^n g_i\frac{\partial}{\partial x_i}$. What I am confused about (this is applied in a homework problem) is when you take a Lie derivative of a Lie derivative i.e. $L_{\textit{v}_1}(L_{\textit{v}_2}f)$. Does this mean, if $\textit{v}_1 = \displaystyle \sum_{i=1}^n g_i\frac{\partial}{\partial x_i}$ and $\textit{v}_2 = \displaystyle \sum_{j=1}^n h_j\frac{\partial}{\partial x_j}$, $L_{\textit{v}_1}(L_{\textit{v}_2}f) = \displaystyle \sum_{i=1}^n g_i\left(\displaystyle \sum_{j=1}^n h_j\frac{\partial^2 f}{\partial x_i \partial x_j}\right)$ or is it something different? Any help would be appreciated!