Derivative of $F(X)G(X)$

Question

Let $X \in \mathbb{R}^{n \times n}$,$F(X) \in \mathbb{R}^{n \times n}$,$G(X) \in \mathbb{R}^{n \times n}$.Does the following expression holds:$\frac{\partial F(X)G(X)}{\partial X}$=$\frac{\partial F(X)}{\partial X}$$(G(X) \otimes I)$+$\frac{\partial G(X)}{\partial X}(I \otimes F(X))$? I have already known there exists such an expression $\frac{\partial F(X)G(X)}{\partial (\vec X)^T}$=$(G(X)^T \otimes I)$$\frac{\partial F(X)}{\partial (\vec X)^T}$+$(I \otimes F(X))\frac{\partial G(X)}{\partial (\vec X)^T}$(Ref: https://www.janmagnus.nl/papers/JRM012.pdf theorem 9).Are they equivalent?Also what are the differences between $\frac{\partial}{\partial (\vec X)^T}$ and $\frac{\partial}{\partial (\overrightarrow{ X^T})}$