Differential Calculus: From Practice to Theory

So far we’ve focused on the natural exponential and logarithm functions. But we’ve seen that \(2^x\) and \(10^x\) are also exponentials, with inverses \(\log_2(x)\) and \(\log_{10}(x) \text{.}\) In fact, we’ve seen that for any \(a\gt 0\text{,}\) \(a\neq1\text{,}\) \(a^x\) is the “base \(a\)” exponential and the corresponding inverse function is called the “base \(a\)” logarithm, and is denoted \(\log_a(x)\text{.}\)

Calculus is not needed for the next problem which establishes that all logarithm functions change multiplication into addition (Property 4) and exponentiation into multiplication (Property 6) as we stated (but didn’t prove) earlier.

But if there are so many different exponential and logarithm functions, don’t we need differentiation rules for all of them? If so, what are the derivatives of the other exponentials and logarithms?

The following problem shows an alternative (and equivalent) method if you can’t remember how to rewrite \(a^x\text{.}\)

Taking a logarithm of complicated formulas sometimes allows us to simplify things considerably via the properties of logarithms. It is a nice trick. Keep it in mind.

Just as we computed the derivative of the general exponential, \(a^x\) by re-expressing it as a natural exponential, we will compute the derivative of the general logarithm, \(\log_a(x)\text{,}\) by re-expressing it as a natural logarithm. This is a bit more difficult, but only a bit.

An easy mnemonic for remembering this formula is to simply notice that on both sides of the equation the \(a\) appears lower than the \(x\text{.}\)

As with the differentiation formula for general exponential functions we (the authors) find it easier to remember this conversion than the differentiation formula. But you are welcome to memorize it if you prefer.

It is a well worn truism among research mathematicians that the first solution of a substantial problem is frequently both very hard to understand and very hard to convey to others, even to other mathematicians. We publish our solutions, in part, so that others can look at the problem, consider the approach used, and find simplifications that the original investigator missed.

This is what happened to Napier. Although his original scheme allowed him to convert multiplication to addition it was more complicated than necessary. Shortly after Napier published his work in \(1614\text{,}\) Henry Briggs

¹⁶

https://mathshistory.st-andrews.ac.uk/Biographies/Briggs/

(1561–1630), Professor of Geometry at Gresham College in London, visited Napier and suggested that a few simple changes to his logarithm table would make it more practical. In particular it was Briggs who suggested that since our number system is in base–\(10\text{,}\) creating a table of base-\(10\) logarithms might be more useful.

Napier had had the same idea himself, but was unable to pursue it “on account of ill-health.” As a result, Briggs took on the task of computing tables of base–\(10\) logarithms, collaborating with Napier until the latter’s death two years later. The results of their collaboration are known as the Briggsian or Common Logarithms. Common Logarithms are the base \(10\) logarithms we discussed earlier and were an important computational aid for nearly \(400\) years until the invention of modern computational technologies.

As the mathematical historian Howard Eves put it, the invention of (common) logarithms literally “doubled the life of the astronomer,” because they so drastically reduced the time spent doing arithmetic.

Most modern scientific calculators have both the natural and the common logarithms built in. The “ln” button on a calculator computes natural logarithms while the “log” computes common logarithms.

The discussion above suggests the question: What base did Napier use when he first invented logarithms, before Briggs suggested using base–\(10\text{?}\)

Napier defined his original logarithms as follows. Consider a point \(C\) moving along line segment \(AB\) and a point \(F\) moving along an infinite ray \(DE\) as seen below.

Since Calculus hadn’t been invented yet, Napier used Trigonometry to develop his table of logarithms. The details of his computations are not relevant for us, except to note that he took the length of \(AB\) to be \(10^7\text{,}\) because the best sine tables at the time were accurate to seven decimal places.

Recall that in Comment 4.3.36 we observed that we could not easily extend the Power Rule for Positive, Rational Exponents to \(y=x^\alpha\) when \(\alpha\) is an irrational number because we didn’t have any way to assign meaning to the expression \(x^\alpha \) when \(\alpha{}\) is irrational.

The properties of the natural exponential and the natural logarithm functions allow us to extend the Power Rule to \(x^\alpha \) for any real value of \(\alpha \) and any positive value of \(x\text{.}\) In particular we can assign meaning to \(x^\alpha \) when \(\alpha \) is irrational.

The same trick we just used can make some differentiations much easier to do.

This looks like a lot of work when it’s laid out on the page but it really isn’t. With practice you can differentiate an expression like \(\frac{\sqrt{x^3+2x}\cdot \tan(x)}{(x^5-7)^4}\) in your head as fast as you can write it down. Really. Even if this is not true, would you rather compute this derivative using the Quotient Rule and Product Rule?

Recall that in Problem 4.3.13 you used the Product Rule to compute \(\dx\left[(x+1)(x+2)(x+3)\cdots(x+n)\right]\text{.}\) You should redo this problem before attempting Problem 8.12.16.

The technique of taking the logarithm of both sides of an expression like \(y=y(x)\) and then differentiating is called Logarithmic Differentiation. It can reduce the amount of tedious computation needed considerably, so it is worth knowing how to use it.

There are some situations where Logarithmic Differentiation is your only option. Consider something like \(y=x^x.\) Unfortunately, this is not a monomial like \(x^2\) nor is it an exponential like \(2^x\text{.}\) This is some strange combination of both and our existing rules don’t directly apply. But Logarithmic Differentiation will work.

Recall that in Chapter 4 we stated, but did not prove the General Power Rule 4.3.37 because we had no way to address the case of an irrational exponent.