Let $\Phi$ be a affine linear mapping with $\Phi(K)=\hat K$, where $\hat K$ is the unit simplex and $\Phi$ is of the form $$\Phi(\hat x)=A\hat x+b$$ with $A\in\mathbb R^{d\times d}$ and $b\in\mathbb R^d$.
How can I determine $\hat D\Phi(\hat x)$, the Jacobian matrix of $\Phi$ at the point $\hat x$?
The Jacobian matrix of an affine linear mapping is the matrix of this mapping, hence $\hat D = A$. It does not depend on the point $\hat x$.