Chain rule for multivariable differentiation

Question

For one-dimensional case, after affine transformation \begin{equation} x=\hat{x}+\sigma c . \end{equation} Using the chain rule for differentiation, we can write \begin{equation} \frac{d}{dc}=\frac{dx}{dc}\frac{d}{dx}=\sigma\frac{d}{dx} \end{equation} \begin{align} \frac{d}{dc}\bigg(\frac{d}{dc}\bigg)&=\frac{d}{dc}\bigg(\sigma\frac{d}{dx}\bigg) \end{align} \begin{align} \frac{d^2}{dc^2}&=\sigma\frac{d}{dc}\bigg(\frac{d}{dx}\bigg)\\& =\sigma \frac{dx}{dc}\frac{d}{dx}\bigg(\frac{d}{dx}\bigg) \end{align} Therefore, \begin{equation} \frac{d^2}{dc^2}=\sigma^2 \frac{d^2}{dx^2}. \end{equation} For multi-dimensional case, \begin{equation} x=\hat{x}+D c, \end{equation} where $x,c \in \mathbb{R}^n$ and $D$ is any lower traiangular matrix of order $n\times n$. How to use chain rule for differentiation \begin{equation} \sum_{i=1}^n\frac{\partial }{\partial c_i}=\text{constant1} \sum_{i=1}^n\frac{\partial}{\partial x_i} \end{equation} \begin{equation} \sum_{i=1}^n\frac{\partial^2}{\partial c_i^2}=\text{constant2} \sum_{i=1}^n\frac{\partial^2}{\partial x_i^2} \end{equation} \begin{equation} \sum_{i=1}^{n} \sum_{j=1 \atop j \neq i}^{n} \frac{\partial ^3}{\partial c_i^2\, \partial c_j} = \text{constant3} \sum_{i=1}^{n} \sum_{j=1 \atop j \neq i}^{n} \frac{\partial ^3}{\partial x_i^2\, \partial x_j} \end{equation} and what will be the values of these constants.