If we consider a damper (dashpot) element that exerts a force opposite the direction of motion proportional to the velocity, i.e. $$ \vec{F} = -c \vec{v}$$

Therefore, we can consider an infinitesimal amount of work done on the damper fluid, $$ dW = F \, dx = (c \dot{x}) \, dx $$

And the time rate of work done on the damper fluid to be, by dividing by $dt$, $$ \frac{dW}{dt} = c \dot{x} \frac{dx}{dt} = c \dot{x}^2 $$

While this makes sense (and can be confirmed within any Dynamics Textbook), I'm uncomfortable with just "dividing by $dt$". How is this legitimate? Is there an alternative way to derive this same thing?