derivative of Pfaffian of (linear) matrix function

Question

Suppose $A: \mathbb{R} \to \mathbb{R}^{2n\times 2n}$ is a single-variate matrix function, with $A(x)$ antisymmetric for all $x$. Wikipedia (https://en.wikipedia.org/wiki/Pfaffian#Derivative_identities) has the identity $$\frac{1}{\mathrm{pf}(A)}\frac{\partial \mathrm{pf}(A)}{\partial x} = \frac{1}{2}\mathrm{tr}\left(A^{-1}\frac{\partial A}{\partial x}\right)$$ for the derivative of the Pfaffian of $A$, which seems to follow from $\mathrm{pf}(A)^2 = \det(A)$ and Jacobi's formula. However, this only applies for values of $x$ such that $A(x)$ is invertible. Is there some kind of extension of this identity that applies even at singular $A(x)$? If it helps, I'm specifically interested in the special case where $A$ is a linear function of $x$ (i.e., $A = B + xC$ for some $x$-independent matrices $B$ and $C$).