\displaystyle\begin{aligned}  \left(\frac{\partial \det X}{\partial X}\right)_{ij}&=\frac{\partial \det X}{\partial x_{ij}}=\frac{\partial\sum_{k}x_{ik}c_{ik}}{\partial x_{ij}}\\  &=\sum_k\delta_{jk}c_{ik}=c_{ij}=\left((\hbox{adj}X)^T\right)_{ij}\\  &=\left((\det X)(X^{-1})^T\right)_{ij}.\end{aligned}