Consider a function $A: \mathbb{R} \rightarrow \mathbb{R}^{n \times n}, t \rightarrow A(t)$. I want to compute the derivative of the determinant
$$\frac{d}{dt} \det(A(t)) \; .$$
Suppose $A(t)$ is invertible at some $t = t_0$, then a closed form solution exists (cf. e.g. The matrix cookbook, p.7),
$$\frac{d}{dt} \det(A(t)) \vert_{t=t_0} = \det(A(t_0)) \;\mathrm{tr}\left(A(t_0)^{-1} \frac{d A(t)}{dt}\vert_{t=t_0}\right) \;.$$
Now, what happens if $A(t_0)$ is not invertible? Is there still a closed form solution?
