3.6: Derivatives of Logarithmic Functions
This section has great because it contains two of my favorite things: a useful tool and a cute proof. Let’s dig in.
First, we start off with a proof of the general case of derivatives for any log function. The proof is one of those great proofs where you mutter “well played Professor Stewart” after reading it. It’s short, simple, clear, and builds upon a number of things you learned in earlier chapters.
You’ll notice that the derivative of is proprtionate to itself multiplied by ln(a). You may be saying to yourself “where the hell did ln come from? That seems like magic.”
Recall from earlier that the derivative of an exponential function is always equal to itself times some constant. Recall in addition that we (we being several centuries of mathematicians plus you and me) agreed that if we found the right base, the proportionality constant would equal 1, and then we called that base e. This amazing and amazingly convenient number was used to determine the character of the constant. Because of that, it ends up being log-base-e of a. It could have been defined by some other more convoluted method, but the cool thing about e is that it simplifies things. So, the take home here is this – as usual, the e is not there by magic. It’s by convention. Mind you, it’s not an arbitrary convention any more than having 2 headlights on a car is arbitrary. It’s a human choice, but it’s an informed smart choice.
Make sense? Sweet.
So, the cool thing is that, since it’s defined in this ln(a) form, as long as the base is e (that is, a=e) shit gets super simple. The derivative of ln(x) is just 1/x.
The examples that follow are, I think, pretty simple. But, there are two cool lessons worth noting.
1) (ln(something))’ = (something)’/(something)
This is fairly obvious, but worth remembering since it helps mental math.
2) (ln|x|)’ = 1/x
This one’s pretty amazing. I say that because absolute value can often be a real bitch. This gives you a nice way to deal with it.
The book makes this one seem a little more complex than it is, in my opinion. That said, it’s a really cool concept.
Here’s the basic idea: say you have an equation of the form
y = (some big ugly combination of exponential expressions)
Using the magic of natural log, you can convert that to
ln(y) = (some big ugly combination of polynomials inside natural logs)
This may make things easier to deal with, since you can just do an implicit differentiation, which converts the left side to y’/y. In other words, ln(y) is about as easy to deal with as y alone. So, if the ln will simplify the right side, it might be a good move. In example 7, you could do the product rule a couple times to expand it out and keep things simple, but that’d be really ugly. By using the log technique, things are… well… not pretty, but a lot less hideous.
Example 8 is even better. It takes a problem that seems pretty intractable and converts it to about 3 lines of simple math. Okay, so the result is surprisingly ugly, but you got what you needed.
Next stop: e as a limit.