From calculus, one obtains the following formula (Leibnitz rule) for the time derivative of an integral that contains time in both the integrand and the limits of the integration:

\[ \frac{d}{d t} \int_{x_{1}(t)}^{x_{2}(t)} f(x, t) d x=\int_{x_{1}}^{x_{2}} \frac{\partial f}{\partial t} d x+f\left(x_{2}, t\right) \frac{d x_{2}}{d t}-f\left(x_{1}, t\right) \frac{d x_{1}}{d t} \]

Discuss how this formula is related to the time derivative of the total amount of a property in a system and to the Reynolds transport theorem.