Answer
$A'(t)=\begin{bmatrix}
-2e^{-2t}
\cos t
\end{bmatrix}$
Work Step by Step
Given: $A(t)=\begin{bmatrix}
e^{-2t}\\
\sin t
\end{bmatrix}$
The derivative of the matrix function is given by:
$\frac{dA(t)}{dt}=\frac{da_{ij}(t)}{dt}$
Hence here, $A'(t)=\begin{bmatrix}
-2e^{-2t}
\cos t
\end{bmatrix}$
