Derivative of determinant of a matrix I would like to know how to calculate:
$$\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big).$$
 A: 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.
A: Think I can provide a proof for Matias' formula.
So, let
$$
A(t) = \mathrm{det}\left( A_1(t), \dots , A_n(t) \right) \ .
$$
By definition,
$$
\frac{dA(t)}{dt} = \mathrm{lim}_{h\rightarrow 0} \frac{A(t+h) - A(t)}{h} = \mathrm{lim}_{h\rightarrow 0} \frac{\det (A_1(t+h), \dots, A_n(t+h)) - \det(A_1(t), \dots , A_n(t))}{h} 
$$
Now, we subtract and add  
$$
\det(A_1(t), A_2(t+h), \dots , A_n(t+h))
$$
obtaining:
$$
\frac{dA(t)}{dt} = \mathrm{lim}_{h\rightarrow 0} \frac{\det (A_1(t+h), A_2(t+h),\dots, A_n(t+h)) - \det(A_1(t), A_2(t+h), \dots , A_n(t+h))}{h} + 
\mathrm{lim}_{h\rightarrow 0}\frac{
\det(A_1(t), A_2(t+h), \dots , A_n(t+h))-\det(A_1(t), \dots , A_n(t))}{h}
$$
Now we focus on the first addend, which is
$$
\det \left( \mathrm{lim}_{h\rightarrow 0} \frac{A_1(t+h) - A_1(t)}{h}, \mathrm{lim}_{h\rightarrow 0} A_2(t+h), \dots,\mathrm{lim}_{h\rightarrow 0} A_n(t+h) \right) 
$$
That is,
$$
\det (A_1'(t), A_2(t), \dots , A_n(t)) \ .
$$
Now, let's go for the second addend to which we substract and add 
$$
\det(A_1(t), A_2(t), A_3(t+h), \dots , A_n(t+h)) \ .
$$
From which we will obtain the term
$$
\det (A_1(t), A'_2(t), A_3(t), \dots , A_n(t)) \ .
$$
Keep on doing analogous operations till you get
$$
\det (A_1(t), A_2(t), \dots , A_{n-1}(t), A_n'(t)) \ .
$$
A: The formula is $$d(\det(m))=\det(m)Tr(m^{-1}dm)$$ where $dm$ is the matrix with $dm_{ij}$ in the entires. The derivation is based on Cramer's rule, that $m^{-1}=\frac{Adj(m)}{\det(m)}$. It is useful in old-fashioned differential geometry involving principal bundles. 
I noticed Terence Tao posted a nice blog entry on it. So I probably do not need to explain more at here. 
A: Like product rule:
$$\dfrac{d}{dt}\det(A_1(t),A_2(t),...,A_n(t))=\det(A_1^{'}(t),A_2(t),A_n(t))+\det(A_1(t),A_2^{'}(t),...,A_n(t))+...+\det(A_1(t),A_2(t),...,A_n^{'}(t))                                  $$
