I have a dynamical system $\dot x(t) = f(x(t))$ with state $x(t)$ taking values in $R^n$ ($t$ is time here). I'm considering the case in which $x(t)$ is a RV with time-dependent PDF $\rho(x,t)$. For a smooth test function $\psi(x)$ with compact support, we look at the expectation $\mu(t) := \int_{R^n} \psi(x) \rho(x,t) dx$. At the beginning of Section 2 of this paper, the author derives the equation of continuity (the Liouville PDE) via the following consideration.
On the one hand, the time derivative $\dot \mu(t)$ can be expressed as $\int_{R^n} \psi(x) \frac{\partial \rho(x,t)}{\partial t} dx$ (OK, Leibniz rule). On the other, noticing that $\frac{d \psi(x(t))}{dt} = \frac{\partial \psi(x(t))}{\partial x} f(x(t))$ (OK, chain rule), we can also write $$\dot \mu(t) = \int_{R^n} \frac{d \psi(x(t))}{dt} \rho(x,t) dx = \int_{R^n} \frac{\partial \psi(x(t))}{\partial x} f(x(t)) \rho(x,t) dx.$$
I don't fully see why the first equality in the latter equation must hold. Any hint on how to prove it?
Thanks for your help!