# Calculus! #34: Early Transcendentals 3.2B

The Quotient Rule

In the last section I broke from form a little by actually working out the proof step by step. I may do this in the future again, but for this section I’ll leave it to the book.

Using similar reasoning to last time, we can come up with a way to deal with derivatives of two functions of the same variable.

Here’s the rule for quotients:

$(\dfrac{f}{g})' = \dfrac{gf' - g'f}{g^2}$

Now, let me tell you why I don’t like the quotient rule.

First off, a quotient is just a different version of a product. f/g can always be changed to f*h, where h=1/g. It might strike you that handling 1/sin(x) is difficult, but after we do the chain rule in 3.4, it’ll seem easy.

Second, with all that in mind, the quotient rule is kind of a bitch to memorize.

Whereas it’s not typically useful, AND it’s a pain to memorize, AND it’s more a tool (as opposed to a deep mathematical truth), I say fuck it. If anyone can tell me why I should not say so, please explain it to me in the comments.

Thus, in my ever so humble opinion, you should only bother with the quotient rule if you’re going to need it for exams. Otherwise, the chain rule is faster and easier.

Next stop: Trig.

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### 7 Responses to Calculus! #34: Early Transcendentals 3.2B

1. J says:

You should learn the quotient rule, because you can about sing it to the tune of Carmen (but only if you use Leibnitz notation).

2. Mike D. says:

Yeah, it’s really easy to swap the gf’ and g’f terms and botch the whole thing.

Another place that h=1/g comes in handy is intensive computations. If you need to normalize a large vector, normally you would just divide every element by N. Thing is, it’s faster to just computer 1/N and multiply every element by that. Floating point (and fixed point) hardware is a *lot* faster multiplying than dividing, sometimes by order N, where N is the number of bits in each element.

3. I agree completely! If you find yourself converting quotients to products, using the product and chain rule, and making common denominators over and over again, then maybe you should memorize this to save yourself a little time. Otherwise, why bother?

Same with the quadratic formula: complete the square and get your answer. And if you’re doing so many of them that you get tired of completing squares, then memorize the result.

4. Chris says:

I completely agree that not typically useful + pain to memorize + tool only => this is a waste of time. However, I’d argue that the quotient rule IS useful in a number of situations. For example, say you want to maximize f(x)/g(x). Since we’re setting the derivative to zero, we can ignore the denominator of the quotient rule and just solve gf’ = fg’. This skips a few steps compared to just using the product rule and chain rule. Part of my proof that this type of situation occurs often is that I still have the quotient rule memorized after ~8 years, so it must be getting used somewhat regularly by my brain.

5. digitrev says:

AUGH! Use \left( and \right). It’ll get those brackets lined up. Please.

6. Ferrett says:

The quotient rule is easy to remember with this mnemonic: lo-de-hi minus hi-de-lo over lo squared. (The “de” is for derivative).

7. Does anyone know a torrent or a file i can download for Essential Calculus Early Transcendentals solution manual, it’s a blue book and believe it’s 1st edition? The solution manual has a red strip over the took. Can anyone help me, pleaseeeee.