Suppose $ A(p)$ and $B(p) $ are functions which map $ \mathbb{R}^{n\times m} $ to $ \mathbb{R}^{n\times m} $ and $$F(p)=S(I_{q}\otimes A(p))(I_{q}\otimes B(p))M$$ where $ S,M $ are constant matrices with dimension $ qn\times qn $ and $ qm\times qm $ respectively. How to calculate $$\frac{d\text{vec}(F(p))}{d\text{vec}(p))}$$
Thank you very much!
Let $\,C=AB\,$ and note that $$(I\otimes A)(I\otimes B)= I\otimes C$$ Let's also use the convention where an uppercase letter denotes a matrix and a lowercase letter a vector, which are related by vectorization, e.g. $$a={\rm vec}(A),\,\,\,b={\rm vec}(B),\,\,\,c={\rm vec}(C),\,\,\,etc$$ Finally, we'll need the SVD of the rightmost matrix in your function. $$M = \sum_k \sigma_ku_kv_k^T$$ Break the function into components (corresponding to an SVD component). $$\eqalign{ F_k &= S(I\otimes C)u_kv_k^T\sigma_k \cr &= S\,{\rm vec}(CU_k)\,v_k^T\sigma_k \cr &= S(U_k^T\otimes I)\,cv_k^T\sigma_k \cr f_k &= \Big(\sigma_kv_k\otimes\big(S(U_k^T\otimes I)\big)\Big)\,c \cr }$$ Now find the derivative of this (vectorized) component matrix. $$\eqalign{ \frac{\partial f_k}{\partial p} &= \Big(\sigma_kv_k\otimes\big(S(U_k^T\otimes I)\big)\Big)\frac{\partial c}{\partial p} \cr }$$ And since $F = \sum_k F_k\,$ we have our answer $$\eqalign{ \frac{\partial f}{\partial p} &= \sum_k\Big(\sigma_kv_k\otimes\big(S(U_k^T\otimes I)\big)\Big)\frac{\partial c}{\partial p} }$$ I'll leave it to you to work out the derivative on the far RHS, seeing as you told us nothing about the functional form of $A(P)$ and $B(P)$ and therefore of $C(P)$.