gradient of nonlinear function

Question

Why is the yellow equality holds ? According to which formulas, meaning, observations ?

Answer 1

Just notice that $$a\in \mathbb{R}^n \Rightarrow dX^Ta \in \mathbb{R}^m \Rightarrow \Phi' dX^Ta \in \mathbb{R}^m \Rightarrow c^T\Phi' dX^Ta \in \mathbb{R} $$ and the trace of a constant is the constant.

To see that $\Phi' : \mathbb{R}^m \to \mathbb{R}^m$ just notice that $\varphi$ takes points of $\mathbb{R}^m$ and can be multiplied at the left by $c^T$.