The left- and right-hand sides of the conservation-of-mass equation
$\frac{\partial c}{\partial t} = - \frac{\partial q}{\partial x}$
are assumed to be everywhere-continuous functions of spatial coordinate $x$ and time $t$, with $c$ being the solute mass concentration (density).
The equation expressing "Fick's Principle" can be derived by fixing $t$ and integrating both sides of the above equation with respect to $x$ over a finite interval, say $[x_1, x_2]$:
$\int_{x_1}^{x_2}\frac{\partial c}{\partial t}\, dx = - \int_{x_1}^{x_2}\frac{\partial q}{\partial x}\, dx$
$\frac{\partial}{\partial t} \int_{x_1}^{x_2}c\, dx = -(q(x_2,t) - q(x_1,t))$
$\frac{\partial }{\partial t}m = q_{in} - q_{out}$
where $m$ is the solute mass in the interval at time $t$.
For similar derivations in the rather more-complicated case of three spatial dimensions, see
continuity equation.