Relationship between the inverse of a matrix function $A(x)$ and the inverse of the derivative of that matrix $\frac{dA}{dx}$

Question

Given an arbitrary matrix function $A(x)$ of dimension $3 \times 3$.

Is there any relationship between the inverse of this matrix and the inverse of the derivative of this matrix, which would allow one to judge the invertibility of the derivative of this matrix by the invertibility of the original matrix?

$A(x)^{-1} \longleftrightarrow [\frac{dA(x)}{dx}]^{-1}$