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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.
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 $\Delta y$.