Most of us learned (and many of us have taught) that the
derivative $dy/dx$ is not the ratio of two quantities dy and dx, but
an indivisible package. The formula $dy/dx = dy/du \cdot du/dx$ is
suggestive but nothing more. Perhaps later we learned that it is
permissible to think of $dy$ and $dx$ as `infinitesimals,' as long as we
understand that this is only a guide to the intuition and cannot be
used for a formal proof. And yet these illegitimate infinitesimals
seem much closer to the spirit of the subject than the awkward
epsilon-delta arguments we go through to avoid them. In this talk we
will explore the legitimate use of infinitesimals, and ponder how they
might make our understanding (and teaching) clearer.