How Much Mathematics Should A Student Memorize? Part 6, Derivatives Of Exponential And Logarithmic Functions

In teaching calculus many, many times over the years, I strove to present to my students my approach to learning and mastering the subject. Part of this approach can be summarized by the slogan

memorize the minimum

As a teacher, I took it as part of my responsibility to help students identify the essential core of material that must be memorized, and then to help students see how they could cope with the rest of the enormous amount of material by relating it to the essential core.

My original motivation for this approach is not very noble: I was a very lazy student, and I hated memorizing things, so I preferred to practice “playing around” with the material so that I could learn tricks to avoid memorizing. I suppose that this also reflects my natural love of sports, games, and play in general … I would much rather be playing than working!

After many years of teaching and learning, and reflecting on both, I realize that although my original motivation may not have been high-minded, it represents good practice. “Knowledge keeps no better than fish,” said Alfred North Whitehead, which I am fond of paraphrasing as follows:

The more you understand, the less you have to memorize.

In calculus class, I would often present two ways to solve a problem, the main method which depends on practicing an essential technique and requires minimal memorization of formulas, and an alternative method that requires simply memorizing and applying a formula. Once I presented the two methods, I would poll the class to see who preferred which method. Invariably, 60%-80% would prefer the method that required the formula.

I understand why students would feel this way: Most of them had little interest in mathematics, were under enormous pressure, and were simply trying to survive with minimal effort. Survival in a world of high-stakes exams often means placing stimulus-response items in short-term memory, to be pulled out on the exam, and permanently forgotten thereafter. They reckoned that this is less work than having to put in the hard work of thinking in order to understand the material. This is a symptom of our crazy education system, but that is a subject for another day. Suffice it to say that I never blamed my students, but I also recognize that for someone who truly desires to understand the subject, the short-term strategy is counterproductive.

Here are a couple of examples of what I mean. I remember the easiest formulas for the derivatives of exponential and logarithmic functions, which I consider part of the essential core:

$\dfrac{{\rm d}}{{\rm d}x} e^x = e^x$

$\dfrac{{\rm d}}{{\rm d}x} \ln x = \dfrac{1}{x}$

What about the other exponential and logarithmic functions? I don’t happen to have these formulas memorized, as I rarely have the need to use them, and have not made a special effort to memorize them. I consider them peripheral, and I would much rather have students practice the following methods for deriving them when needed. In practicing the derivations, students naturally also reinforce the connections between logarithmic and exponential functions, and the means for transforming between them.

Suppose you wish to determine the derivative of the exponential function $y = b^x$, where $b$ is some positive number. Strategy: Take the natural logarithm of both sides, simplify the right side using properties of logarithms, differentiate implicitly, and then solve for $y^{\prime}$:

$y = b^x $
$\ln y = \ln \left ( b^x \right ) $
$\ln y = x \ln b $
$\dfrac{1}{y} \cdot y^{\prime} = \ln b $
$y^{\prime} = \left ( \ln b \right ) y $
$\dfrac{{\rm d}}{{\rm d}x} b^x = \left ( \ln b \right ) b^x $

On the other hand, suppose you wish to differentiate the function $y = \log_b x$. The strategy here is to switch to exponent form, take the natural logarithm of both sides, and continue as above:

$y = \log_b x $
$b^y = x $
$\ln b^y = \ln x $
$y \ln b = \ln x $
$y^{\prime} \cdot \ln b = \dfrac{1}{x} $
$\dfrac{{\rm d}}{{\rm d}x} \log_b x = \dfrac{1}{\ln b} \cdot \dfrac{1}{x} $

With practice, one can perform these calculations in seconds. And in practicing them one deepens and demonstrates one’s understanding of many essential process skills.

If you seek deep understanding, don’t memorize the results of these two calculations! Practice the calculations instead.

* * *

Earlier posts in this series:

How Much Mathematics Should a Student Memorize? Part 5, The Multiplication Table

How Much Mathematics Should a Student Memorize? Part 4, Geometric Series

How Much Mathematics Should a Student Memorize? Part 3, The Graphs of Power Functions

How Much Mathematics Should a Student Memorize? Part 2, Integral Calculus

How Much Mathematics Should a Student Memorize? (Part 1, Trigonometric Identities)

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 21 January 2021.)