If $T : \mathbb{R}^n \to \mathbb{R}$ is multilinear then $$ \frac{d}{dt}T(x_{1}(t),\dots,x_{n}(t)) = T(\dot{x}_{1}(t),\dots,x_{n}(t)) + \cdots{} + T(x_{1}(t),\dots,\dot{x}_{n}(t)) $$ Prove that, if $X(t)= (x_{1}(t), \dots, x_{n}(t))$ with every $x_i : \mathbb{R} \to \mathbb{R^n}$ and $X(t_{0})=I_{n \times n}$, then $$ \frac{d}{dt} \det(X)(t_{0}) = \mbox{tr} \left( \dot{X}(t_{0}) \right) $$
I got stuck in this problem for my introduction to PDE course. Any hint?