I was looking for theorems that might be helpful in order for some proofs that I have and I came across the following one:

$$\frac{d}{dt} [\det A(t)]=\det A(t) \cdot \operatorname*{tr}[A^{-1}(t)\cdot \frac{d}{dt} A(t)]$$

where $A(t)$ is a matrix with a variable $t$.

The problem is that I have neither a reliable source for this theorem nor am I able to prove it.

Did anyone come across the aforementioned equation or is able to prove it?