How to calculate the impulse response of an RC circuit using time-domain method

I have an $RC$ circuit and I calculate the differential equation as
$$RC \frac{dV_c}{ dt} + V_c = V_g$$ that is the same of
$$RCy'(t) + y(t) = x'(t).$$ If I assume $x(t) = u(t)$ I have that the equation is $$RCg’(t) + g(t) = 0$$ and the solution is $$g(t) = K_1 e^{-t/RC} + K_0$$ now for have the impulse response I have to calculate this for $t > 0$, $t < 0$ and $t= 0$ but I have no idea how to do. Can you please help me ?