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The first derivative is $\dot{x}=Ae^{-t}(1-t)$, and the second derivative is $\ddot{x}=Ae^{-t}(t-2)$. These are computed by applying the product rule successively to the trial solution $x=Ate^{-t}$.

Conditions: The trial solution is $x=Ate^{-t}$.; Differentiation is with respect to $t$.; The product rule is applied to $t$ and $e^{-t}$.

The first derivative is $\dot{x}=Ae^{-t}(1-t)$, and the second derivative is $\ddot{x}=Ae^{-t}(t-2)$. These are computed by applying the product rule successively to the trial solution $x=Ate^{-t}$.

Conditions: The trial solution is $x=Ate^{-t}$.; Differentiation is with respect to $t$.; The product rule is applied to $t$ and $e^{-t}$.