Recall the definition of ordinary differentiation. In light of the
treatment above, we now recognize that the ``ordinary'' differentiation
we learned in the first year of calculus was ordinary because it was
*scalar* differentiation - differentiation of functions that
represent scalar quantities. Given a (continuous, differentiable - we
will assume this unless stated otherwise for all functions discussed)
function
:

Note my explicit and deliberate use of
as the independent variable
upon which
depends. This invites us to think of this as a
*rate of change* in physics where
is some physical quantity as
a function of
the *time*.

From this one can easily derive all sorts of associated rules, the most important of which are:

- The Chain rule. Suppose we have a function
where
is itself a function of
(and there is no ``separate'' time
dependence in
). Then:
- The Sum rule. Suppose we have two functions,
and
. Then:
- The Product rule. Suppose we have two functions,
and
. Then:

We will often express these rules in terms of *differentials*, not
*derivatives with respect to specific coordinates*. For example:

Most of these simple scalar rules have counterparts when we consider different kinds of vector differentiation.