Let $ A= (a_{i,j})$ be a $d \times d$ matrix whose entries are real functions of $t$ Then,

$$\begin{split} \frac{d^n}{dt^n}\det(A) &=\sum_{n_1+\cdots+n_d=n} {n \choose n_1,n_2,...,n_d} \sum_{\pi \in S_d} \sgn(\pi) \prod_{i=1}^{d} \frac{d^{n_i}}{dt^{n_i}}a_{i,\pi(i)}\\ \\ &= \sum_{n_1+\cdots+n_d=n} {n \choose n_1,n_2,...,n_d} \det \begin{pmatrix} \frac{ d^{n_1}}{dt^{n_1}}a_{1,1} & \frac{ d^{n_1}}{dt^{n_1}}a_{1,2} & \cdots &\frac{d^{n_1}}{dt^{n_1}}a_{1,d} \cr \vdots & \vdots & & \vdots \cr \frac{d^{n_d}}{dt^{n_d}}a_{d,1} & \frac{d^{n_d}}{dt^{n_d}}a_{d,2} & \cdots & \frac{d^{n_d}}{dt^{n_d}}a_{d,d} \end{pmatrix} \end{split}$$

where $ {n \choose n_1,n_2,...,n_r}$ is the multinomial coefficient, $S_d$ is the symmetric group of permutations and $\sgn(\pi)$ is the sign of a permutation $\pi$