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Differentials

We have learned that the symbols  df  and  dx  represent the coupled changes in f(x) and x, in the limit where the change in x (and consequently also the change in f) become infinitesimally small. We call these symbols the differentials of f and x and distinguish them from $\Delta f$ and $\Delta x$ only in this sense: $\Delta f$ and $\Delta x$ can be any size, but df and dx are always infinitesimal - i.e. small enough so that we can treat f(x) as a straight line over an interval only dx wide.

This does not change the interpretation of the representation   ${\displaystyle {df \over dx}}$  for the derivative of f(x) with respect to x, but it allows us to think of these differentials df and dx as "normal" algebraic symbols that can be manipulated in the usual fashion. For instance, we can write

$$df = \left( df \over dx \right) dx$$

which looks rather trivial in this form. However, suppose we give the derivative its own name:

$$g(x) \equiv {df \over dx}$$

Then the previous equation reads

$$df = g(x) \; dx \qquad \hbox{\rm or just} \qquad df = g \; dx$$

which can now be read as an expression of the relationship between the two differentials df and dx. Hold that thought.

As an example, consider our familiar kinematical quantities

$$a \equiv {dv \over dt} \qquad \hbox{\rm and} \qquad v \equiv {dx \over dt}.$$

If we treat the differentials as simple algebraic symbols, we can invert the latter definition and write

$${1 \over v} = {dt \over dx}.$$

(Don't worry too much about what this "means" for now.) Then we can multiply the left side of the definition of a by 1/v and multiply the right side by dt/dx and get an equally valid equation:

$${a \over v} = {dv \over dt} \cdot {dt \over dx} = {dv \over dx}$$

or, multiplying both sides by  $v \, dx$,

$$a \; dx \; = \; v \; dv$$ (11.1)

which is a good example of a mathematical identity, in this case involving the differentials of distance and velocity. Hold that thought.