Jacobi's formula

Remarks

The determinant is a polynomial in the entries $a_{ij}$. By the Laplace expansion along row $i$, its partial derivative with respect to $a_{ij}$ is the cofactor $C_{ij}$. The chain rule gives $\frac{d}{dt}\det A = \sum_{i,j} C_{ij}\, a_{ij}'$. This sum is $\tr(\operatorname{adj}(A) A')$, because $\operatorname{adj}(A)_{ji} = C_{ij}$. The second form follows from $\operatorname{adj}(A) = \det(A) A^{-1}$.

A special case is $\frac{d}{dt}\det(I + tX)\big|_{t=0} = \tr X$: the derivative of the determinant at the identity is the trace. For $A(t) = e^{tX}$, the formula gives $\frac{d}{dt}\det e^{tX} = \tr(X) \det e^{tX}$, so $\det e^{tX} = e^{t \tr X}$.

Going deeper

For a student

The formula says that the differential of $\det$ at $A$ is the linear map $X \mapsto \tr(\operatorname{adj}(A) X)$. It gives Liouville's formula for the Wronskian of a linear system of differential equations. It also gives the determinant of a matrix exponential, $\det e^A = e^{\tr A}$. In Lie theory it says that the derivative of $\det \colon \mathrm{GL}_n(\mathbb{K}) \to \mathbb{K}^\times$ at the identity is the trace.