4.8.1 Background and a Calculation

It's time to answer the question, "What's a differential?" Perhaps you have forgotten the question. It came up in Chapter 2 when we introduced the notation `dytext[/]dt` for the instantaneous rate of change of `y` with respect to `t`. The numerator `dy` and the denominator `dt` were called differentials to justify the standard name "differential equation" for an equation containing a derivative of the unknown function. In fact, the portion of calculus that pertains to the study of derivatives is universally known as differential calculus, so we really have an obligation to tell you why.

The concept of differential and the quotient notation for the derivative are part of the contribution to the development of calculus made by Leibniz — a small but very important part. As we will see from time to time, notation can have power. Good notation is a powerful tool enabling us to develop concepts and to solve problems. Bad notation can seriously hinder our efforts to do either.

We have already seen an example of the use of Leibniz notation in the development of a concept: the Chain Rule. The notation immediately suggests both the correct statement of the rule and a means for showing that it is indeed correct. Our other notation for derivatives, the prime notation that traces its heritage to Newton, would not have served us nearly so well.

Note 1 – Leibniz and Newton

To illustrate the power of the Leibniz notation for problem solving, we provide a small problem you can solve now — with some effort — with nothing but elementary geometry and perhaps a calculator. Later, we will show how the problem can be solved with a quick pencil-and-paper calculation that uses Leibniz notation.

Activity 1

Estimate the total volume of the earth's crust. You may assume the earth is a sphere with a radius of `4000` miles, and the average depth of the crust is `20` miles.

Comment on Activity 1

As you may have guessed by now, the concepts of differential and derivative are closely related, but they are not the same thing. To explain the difference, we once again call on the fundamental concept for this course:

This equation is meaningful, of course, only if there is a run. That is, it describes the calculation of average rate of change (slope), but it does not directly explain the calculation of instantaneous rate of change (derivative), except through an approaching process in which the run shrinks to zero.