Video Lecture

In this segment we aim to develop a new technique to handle certain kinds of derivatives; namely derivatives of functions that are products of several factors.

For a look at how to know when (and why) to use logarithmic differentiation, you can check out the supplemental video below!

https://youtu.be/PnhXCJMyMTQ

(Supplemental Videos are included via external link so you don’t have to watch them to earn credit.)

Text and Additional Details

Consider the function $y = \frac {(x+1)^2(x-2)^3(x+3)}{(x-7)(x+5)(x-1)}$. Imagine we wanted to take a derivative of this function. There are a couple options that come to mind:

(a)

First, we could try to take the derivative as it stands, but there are four products and two of them have powers... that’s a lot of product rules. Actually, depending on how we break it apart, it would take at least five applications of product rules, and several chain rules, not to mention a quotient rule and a rather horrifying expansion that would generate an incredible amount of opportunities for computation errors.

(b)

Our Second option is somehow even worse... we could expand all of that out into a one giant polynomial and then take a derivative. Good luck with that!

Unfortunately, this kind of function, in this kind of form, is actually very common. Polynomials are the preferred function of computers, and factored form is endlessly useful, which means most polynomials tend to be in factored form (for other reasons) before you want to take a derivative. Between these two facts, having to take derivatives of large products like this isn’t an especially rare type of problem to run into. Fortunately, there is indeed a better way... implicit differentiation!

Now, you are probably thinking, “implicit differentiation? But it’s an explicitly defined function!” (Or, perhaps “implicit what? Oh right... really though?”) Well, don’t worry, we can always make the problem more complicated (cue maniacal laughter)! Really though, stick with me for a moment. Instead of taking a derivative immediately, let’s start by taking a natural log of both sides first. This gives us the following;

So, aside from some kind of innate massochism, why do this? Because log rules will pull us out of this headache of a situation.

Recall that $\ln (ab) = \ln (a) + \ln (b)$ and $\ln (a^b) = b\ln (a)$. We can use these two properties to split our expression into the following;

Now we can easily apply our derivative. The right hand side may still look terrible, but it actually is quite straightforward; recall that $\frac {d}{dx} \ln (u) = \frac {1}{u} \cdot u'$.

So we have:

Now, on the one hand, our result looks pretty ugly... it isn’t factored very well, nor is it particularly compact. But the derivative step was relatively painless. As always there are pros and cons to any given technique. The “pro” of this technique is that we can abuse the nice way derivatives interact with exponentials and logarithms, along with log rules, to make the derivative process much easier. The “con” is that we end up with an expression that isn’t particularly pretty or pleasant to work with if we need to do anything that involves factoring or something similar.

This technique is called Logarithmic Differentiation, since we are using logarithms to expedite the process of taking the derivative. There is another video (linked at the top of the page below the lecture video) covering how to know when to use this technique, and more importantly, when using it would be a pretty bad idea.

We have seen a fundamentally new approach to derivatives; not simply a rule to make a specific core function easier to differentiate, and not a technique that we’ve seen in precalculus like adding zero or multiplying by one cleverly. This is a technique specifically tailored to taking derivatives, designed to make large factored functions more accessible, at least for the derivative step.