Chapter B, Problem 58E

To determine

To prove: The product rule $\frac{d}{dt}\left[A\left(t\right)X\left(t\right)\right]=A\left(t\right)X^{\prime }\left(t\right)+A^{\prime }\left(t\right)X\left(t\right)$, if $A\left(t\right)$ is a $2\times 2$ matrix of differentiable functions and $X\left(t\right)$ is a $2\times 1$ column matrix of differentiable functions.