Answer
$A'(t)=\begin{bmatrix}
e^t & 2e^{2t} & 2t \\
2e^{t} & 8e^{2t} & 10t\\
\end{bmatrix}$
Work Step by Step
Given: $A(t)=\begin{bmatrix}
e^t & e^{2t} & t^2\\
2e^t & 4e^{2t} & 5t^2
\end{bmatrix}$
The derivative of the matrix function is given by:
$\frac{dA(t)}{dt}=\frac{da_{ij}(t)}{dt}$
Hence here, $A'(t)=\begin{bmatrix}
e^t & 2e^{2t} & 2t \\
2e^{t} & 8e^{2t} & 10t\\
\end{bmatrix}$
