Answer
$A'(t)=\begin{bmatrix}
\cos t & -\sin t & 0 \\
\sin t & -\cos t &1\\
0 & 3 & 0
\end{bmatrix}$
Work Step by Step
Given: $A(t)=\begin{bmatrix}
\sin t & \cos t & 0\\
-\cos t & \sin t & t \\
0&3t&1
\end{bmatrix}$
The derivative of the matrix function is given by:
$\frac{dA(t)}{dt}=\frac{da_{ij}(t)}{dt}$
Hence here, $A'(t)=\begin{bmatrix}
\cos t & -\sin t & 0 \\
\sin t & -\cos t &1\\
0 & 3 & 0
\end{bmatrix}$
