I want to solve part of an energy minimization problem like this: \begin{align*} \quad && \arg \min_{\textbf{X}} \sum_{i=1}^{I} \sum_{i' \in neighnors(i)} \frac{1}{ {\left\| (\textbf{x}^i - \textbf{x}^{i'}) - (\textbf{x}^i - \textbf{x}^{i'})^T\textbf{n}_{i'}\textbf{n}_{i'} \right\|}_{2}^{2} } && (1) \end{align*} where $\textbf{X}$ is the positions of point sets, $I$ is the number of points. $\textbf{x}^i$, $\textbf{x}^{i'}$ and $\textbf{n}_{i'}$ are $3\times1$ vectors.
To use a gradient decent method, I need to solve the derivative like this: \begin{align*} \quad && \frac{ \partial }{ \partial\textbf{x}^{i}} \left( \frac{1}{ {\left\| (\textbf{x}^i - \textbf{x}^{i'}) - (\textbf{x}^i - \textbf{x}^{i'})^T\textbf{n}_{i'}\textbf{n}_{i'} \right\|}_{2}^{2} }\right) && (2)& \end{align*}
I have two main questions:
1.Suppose $g(\textbf{x}^i)={\left\| (\textbf{x}^i - \textbf{x}^{i'}) - (\textbf{x}^i - x^{i'})^T\textbf{n}_{i'}\textbf{n}_{i'} \right\|}_{2}^{2}$, is the chain rule still correct? In other words, is equation (3) correct? \begin{align*} \quad && \frac{\partial \frac{1}{g(\textbf{x}^i)}}{\partial\textbf{x}^{i}} = \frac{\partial \frac{1}{g(\textbf{x}^i)}}{\partial g(\textbf{x}^i)} \bullet \frac{\partial g(\textbf{x}^i)}{\partial\textbf{x}^{i}} && (3)& \end{align*}
2.How to solve this: \begin{align*} \quad && \frac{ \partial ((\textbf{x}^i - \textbf{x}^{i'})^T\textbf{n}_{i'}\textbf{n}_{i'}) }{\partial\textbf{x}^{i}} && (4)& \end{align*}