Using Product rule to take derivative

Question

I'm trying to take the derivative of: $$\frac{-1}{6}(e^{-3t}-1) u(t)$$

The $u(t)$ is the step response. So the answer I get is by just doing product rule: $$\frac{1}{2}e^{-3t}u(t)-\frac{1}{6}e^{-3t}\delta(t)$$ but wolfram gets a different answer.

Why does: $-\frac{1}{6}e^{-3t}\delta(t)$ goes to $0$?

Answer 1

Because "it has a bug"?

In my MMA (v8.0,Linux-x64),

 D[-(1/6) (E^(-3 t) - 1) HeavisideTheta[t], t]

outputs

 -(1/6) (-1 + E^(-3 t)) DiracDelta[t] 
                          + 1/2 E^(-3 t) HeavisideTheta[t]