Another Mathematical Interlude: What is a derivative?

by George E. Hrabovsky, President MAST

A Note to the Reader

You will notice that I ask numerous questions and number them. I have no intention of answering these questions in sequence. The numbering is a bookkeeping device so that I can keep track of things. The point of this column is to investigate what heat is, and I will only go so far afield from that goal. This means that some questions will go unanswered. This is reasonable, and it allows for future projects based on those unanswered questions. Feel free to attempt to answer these questions for yourself.

Where We Have Been

We started to get back to physics last time, only to be stumped by a mathematical question of great importance.

Session 6: What is a derivative?

For this question I will introduce a new source [1] and adopt its take on things.  A derivative is the quotient of two related but very small quantities.  The numerator is the minute change in a function that depends on a specific variable.  The denominator is the minute change in that variable.  We use the symbol to denote this minute change.  We can think of it like this, if we have,

and we want to know what happens to as increases by a minute quantity, , then

This is called the differential of .  Likewise, is the differential of .  These quantities are so small that we can ignore them.

The derivative is the quotient of these quantities.

For example, say that

Then,

The derivative is then,

Since we are treating as so small we can ignore it, then we just treat it as 0.  So,

    Another example is,

So,

Then,

Again,  we are treating as so small we can ignore it, then we just treat it as 0.  So,

So, we now know that acceleration is the time derivative of velocity.

What is velocity?

Velocity is the rate of change of an object's position with respect to time.  The average velocity of an object is given by the formula,

Here is the average velocity, is the average displacement.  For example, assuming we measure the position at two times, and we write these times as and , then the displacement is

Given our discussion of derivatives above, velocity becomes the time derivative of position.

Book Review: Essential Physics

Silvanus Thompson and Martin Gardner, "Calculus Made Easy," St. Martin's Press (1998).

This is a classic work that has been revised at least four times since it came out in 1910.  It has been reprinted many times, and this latest edition has been expanded by Martin Gardner to include three new introductory chapters. The text has been completely rewritten in more modern notation. This hardcover version costs around $21, a rarity for calculus texts. 

This book takes a less rigorous approach to calculus than almost any book around today by explaining things only so far as is needed. Do not use this book to learn the theory behind calculus; use it to learn to do calculus.

Martin Gardner is well known for producing scientific and mathematics books written for lay-people and enthusiasts. The first chapter he added to this book introduces functions and their graphs. The second chapter discusses limits, sequences, and series. His third chapter is all about the slope of a line.

The first real chapter is called, "To Deliver You From The Preliminary Terrors," and as such is all about the notation used in calculus. It sets the tone for the rest of the book, and the book tries very hard not to take itself too seriously.

Chapter 2 discusses differentials, explaining how you can consider that something is so small as to ignore it. Chapter 3 is all about constants and variables. The rate of growth of a linear function is covered, and derivatives are introduced. Chapter 4 establishes the power rule. Chapter 5 establishes the constant rule and the constant multiple rule.  

Chapter 6 establishes the sum rule, the difference rule, the product rule, and the quotient rule. Chapter 7 discusses successive differentiation. Chapter 8 discusses time derivatives. Chapter 9 establishes the chain rule. Chapter 10 explores the geometry of derivatives. This leads to Chapter 11 on maxima and minima of functions, one of the most useful applications of calculus. Chapter 12 returns to geometry, this time speaking of measuring the curvature of a function. Chapter 13 discusses first partial fractions and then inverse functions. Chapter 14 covers series, exponential functions, and logarithmic functions.

Chapter 15 shows how to differentiate trigonometric functions, including the strangeness that happens with successive differentiation of trigonometric functions. Chapter 16 ends the discussion of differentiation by introducing partial derivatives, that is derivatives of functions of more than one variable with respect to one variable.

Chapter 17 covers the idea of integration, and this occupies the remainder of the book. Chapter 18 introduces the idea of integration as the opposite of differentiation. Chapter 19 develops the fundamental theorem of calculus in an almost breathtakingly simple way. Chapter 20 develops such notions as integration by parts, integration by substitution, integration by partial fractions, and introduces the idea of a differential equation; one of the most important developments in mathematics ever!  

Chapter 21 covers how to solve differential equations. Chapter 22 returns to the notion of measuring the curvature of curves. Chapter twenty three discusses how to use integration to determine how far a curve extends. This is followed by a short list of derivatives and integrals. Then there are answers to many of the practice problems found at the end of most chapters. Finally, Martin Gardner concludes with a chapter on recreational mathematics using calculus.

This is a book that mathematicians might not appreciate because of its lack of rigor. Students have loved this book, and have kept it in print, for nearly one hundred years; that should tell you something right there. I understand the desire for mathematicians to build everything on a firm basis before building in it. I myself do not use the approach identified in this book when I teach calculus. But the book always makes me think I am doing something wrong by discussing limits first. All I can say is that I think this a terrific book to learn how to do calculus.

References

[1] Silvanus Thompson and Martin Gardner, "Calculus Made Easy," St. Martin's Press (1998).

Created by Mathematica  (February 28, 2005)