My question can be summarized as: Let $\mathcal{M}$ be a smooth manifold and $T\mathcal{M}$ be its tangent bundle. It's well known that $T\mathcal{M}$ can be viewed as a smooth manifold. Then how does the coordinate transformations look like?
Let $(x^i)$ be local coordinates on $\mathcal{M}$ and for any $v \in T_x(\mathcal{M})$, suppose that $v=v^i\frac{\partial}{\partial x^i}$. Then $(x^i, v^i)$ form coordinates on $T\mathcal{M}$. Let $(x'^a, y'^a)$ be another chart on $T\mathcal{M}$, I thought that there should be a transformation like \begin{equation} \frac{\partial}{\partial x^{\prime a}}=\frac{\partial x^i}{\partial x^{\prime a}} \frac{\partial}{\partial x^i} \end{equation}
However, in this note https://arxiv.org/abs/1810.04257, on page 2, the second footnote, it says \begin{equation} \frac{\partial}{\partial x^{\prime a}}=\frac{\partial x^i}{\partial x^{\prime a}} \frac{\partial}{\partial x^i}+v^{\prime b} \frac{\partial^2 x^j}{\partial x^{\prime a} \partial x^{\prime b}} \frac{\partial}{\partial v^j} \end{equation}
Where does the term $v^{\prime b} \frac{\partial^2 x^j}{\partial x^{\prime a} \partial x^{\prime b}} \frac{\partial}{\partial v^j}$ come from?