A point $P$ moves along the curve $y = \ln(x)$. Let $A$ be the area of the rectangle formed by the origin, the point $(x, 0)$, the point $(x, \ln x)$, and the point $(0, \ln x)$. If $x$ is increasing at a constant rate of $2$ units per second, determine the rate of change of the area $A$ with respect to time $t$ at the instant $x = e$.