I always explain in terms of linear approximation. The "derivative" of $f(x)$ is the function $f'(x)$ for which linear approximation holds, i.e. if we change $x$ to $\Delta x$ then how does $f(x)$ change? $$ f(x+\Delta x) = f(x)+ f'(x)\Delta x + O(\Delta x)^2 $$ The example I give my section students is $100.17^2 \approx 10034$ Do we care about the extra 0.0289? probably not.
Also real world data is not continuous time, so we are always estimating the rate of things.
The infinitesimal point of view is useful in math an physics. One exercise is to check Green's theorem $\oint Pdx + Qdy = \int \int \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \, dx dy$ by integrating on/in an infinitesimal rectangle of width $\Delta x$ and height $\Delta y$.
I also recommend Infinitesimal Calculus by James M. Henle and Eugene M. Kleinberg as a point of view on how to teach Calc I & II
