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In this note, I show that it is possible to use elementary mathematics,instead of the machinery of Lambert function, Laplace Transform, or numerics,to derive the instability condition, $k \tau = \pi/2$, and the critical dampingcondition, $k\tau = 1/e$, for the time-delayed equation $\dot{\theta} = -k\theta(t-\tau)$. I hope it will be useful for the new comers to this equation,and perhaps even to the experts if this is a simpler method compared to otherversions.

On the Instability and Critical Damping Conditions, $k\tau = 1/e$ and $k\tau = \pi/2$ of the equation $\dot{\theta} = -k \theta(t-\tau)$