n’th derivative of a determinant

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

\\begin{split}\\displaystyle\\frac{d^{n}}{dt^{n}}\\det(A)&\\displaystyle=\\sum_{n_{1%}+\\cdots+n_{d}=n}{n\\choose n_{1},n_{2},...,n_{d}}\\sum_{\\pi\\in S_{d}}%\\operatorname{\\mathrm{sgn}}(\\pi)\\prod_{i=1}^{d}\\frac{d^{n_{i}}}{dt^{n_{i}}}a_{%i,\\pi(i)}\\\\\\\\&\\displaystyle=\\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 $\\operatorname{\\mathrm{sgn}}(\\pi)$ is the sign of apermutation $\\pi$ .