Suppose that $\{f_1, \ldots, f_n\}, \{g_1, \ldots, g_n\}$ and $\{h_1, \ldots, h_n\}$ are algebraically independent polynomials that generates the same algebra of $\mathbb{R}[x_1, \ldots, x_n]$. Then I need to probe the following chain rule $$\frac{\partial g_i}{\partial h_j} = \sum_{k = 1}^n \frac{\partial g_i}{\partial f_k} \frac{\partial f_k}{\partial h_j}.$$
I don't have a definition for $\frac{\partial g_i}{\partial h_j}$, but I guest is $\frac{\partial g_i}{\partial h_j} = \frac{\partial G_i}{\partial x_j}$, where $g_i = G_i(h_1, \ldots, h_n)$.
This could be easy, but I need a formal point of view.