As with trigonometric functions, you’ll be expected to remember all of the logarithmic and exponential
functions you’ve studied in the past. If you’re not sure about any of this stuff, review the unit on
Prerequisite Mathematics. Also, this is only part one of our treatment of exponents and logs. Much of what
you need to know about these functions requires knowledge of integrals (the second half of the book), so
we’ll discuss them again later.
THE DERIVATIVE OF ln x
When you studied logs in the past, you probably concentrated on common logs (that is, those with a base
of 10), and avoided natural logarithms (base e) as much as possible. Well, we have bad news for you:
Most of what you’ll see from now on involves natural logs. In fact, common logs almost never show up in
calculus. But that’s okay. All you have to do is memorize a bunch of rules, and you’ll be fine.
Rule No. 1: If y = ln x, then = This rule has a corollary that incorporates the Chain Rule and is actually a more useful rule to memorize.
Rule No. 2: If y = ln u, then = Remember: u is a function of x, and is its derivative.
You’ll see how simple this rule is after we try a few examples.
Example 1: Find the derivative of f(x) = ln(x^3 ).
f′(x) = = If you recall your rules of logarithms, you could have done this another way.
ln(x^3 ) = 3 ln xTherefore, f′(x) = 3.