Solution
= Solution
For the <inflaton>,
$$
\dot\rho=\dot\phi\ddot\phi+V'(\phi)\dot\phi,
\qquad
\rho+P=\dot\phi^2.
$$
The continuity equation therefore becomes
$$
\dot\phi\left[\ddot\phi+3H\dot\phi+V'(\phi)\right]=0.
$$
Away from an isolated turning point division by $\dot\phi$ gives, and continuity extends through such points,
$$
\boxed{\ddot\phi+3H\dot\phi+\frac{dV}{d\phi}=0}.
$$