Answer
$A'(t)=\begin{bmatrix}
1 & \cos t\\
-\sin t & 4
\end{bmatrix}$
Work Step by Step
Given: $A(t)=\begin{bmatrix}
t & \sin t\\
\cos t & 4t
\end{bmatrix}$
The derivative of the matrix function is given by:
$\frac{dA(t)}{dt}=\frac{da_{ij}(t)}{dt}$
Hence here, $A'(t)=\begin{bmatrix}
1 & \cos t\\
-\sin t & 4
\end{bmatrix}$
