This isn't quite the same as the rule above. Suppose . Then:

where is a

In physics (or anything else where the symbols can have units and aren't
just pure numbers), this is *not* generally true unless the
arguments are dimensionless. In fact this is a specific example of a
*general rule* that one cannot substitute equalities that carry
units into any functional form that has a power-series expansion. On
the other hand one *can* substitute in quantities that in some sense
``look the same'' that are *dimensionless*. This is easy to
understand. Supposed I know that is a length in meters. I *can* certainly write down its exponential: . But does this make
sense?

If I expand in its well-known power series (summarized, derived,
defined later):

we see that it is in fact

The only way this can make sense is if we write something like:

where is some