TITLE: Derivative of determinant of a matrix
QUESTION [21 upvotes]: I would like to know how to calculate:
$$\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big).$$

REPLY [4 votes]: In the previous answers it was not explicitly said that there is also the Jacobi's formula to compute the derivative of the determinant of a matrix.
You can find it here well explained: JACOBI'S FORMULA.
And it basically states that:

Where the adj(A) is the adjoint matrix of A. How to compute the adjugate matrix is explained here: ADJUGATE MATRIX.
I hope it will help someone.