Scalar Differentiation

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 $f(t)$ :

$$\ddx{f} = \lim_{\Delta t \to 0} \frac{f(t + \Delta t) - f(t)}{\Delta t}$$

Note my explicit and deliberate use of $t$ as the independent variable upon which $f$ depends. This invites us to think of this as a rate of change in physics where $f$ is some physical quantity as a function of $t$ 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 $f(x)$ where $x(t)$ is itself a function of $t$ (and there is no ``separate'' time dependence in $f$ ). Then:
  $$\ddt{f} = \ddx{f}\ddt{x}$$

• The Sum rule. Suppose we have two functions, $f(t)$ and $g(t)$ . Then:
  $$\ddt{(f+g)} = \ddt{f} + \ddt{g}$$

• The Product rule. Suppose we have two functions, $f(t)$ and $g(t)$ . Then:
  $$\ddt(fg) = g \ddt{f} + f \ddt{g}$$

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

$$df = \ddx{f} dx = \ddt{f} dt$$

$$d(fg) = g\ df + f\ dg$$

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