How is a raven like a writing desk? (Quotient Rule -> Product Rule)
Lewis Carroll gave this line to the Mad Hatter with the intention of it having no answer. Of course, literary critics and others of similar ilk have decided that Carroll was a sod, and that his riddle did indeed have an answer: Poe wrote on both. Perhaps there’s an essay on over-interpretation to be written? English teachers?
You all know the following, but I think it’s important enough to discuss in detail. The quotient rule can be found using the product rule, thus showing its redundancy. I show very few choice proofs and derivations to high school students (which some of you will be up in arms about… and I’ve discussed this before). This is one of them.
Now that the students have learned the chain rule, the equality of the chain rule to the quotient rule opens up to them. I generally start them here:
This gets at negative exponents (which are generally a rough spot for my kids) and arbitrary functions. We reason, that if indeed this fraction is equal to this product, then we ought to be able to use the product rule to find the derivative of the fraction by working on the left-hand side. Doing this will require us to use the chain rule within an implementation of the product rule( which slides me directly into my next standard: mixed differentiation rules) So, apply d/dx:
Avoid the quotient rule altogether:
The first term is the derivative of the first function with the second left as is. The second term is the first function left as is and the chain rule of the second function. Let’s clean that up:
Now, another student favorite, common denominators:
and, finally:
Thus, the quotient rule is shown to be a resulting artifact of the product and chain rules. My students used this to check their work. They also use this understanding to better algebraically manipulate functions to choose the simplest way to find their derivatives, which is what every calculus teacher secretly wants: students showing understanding of efficiency and elegance in mathematics.
And now, Spring Break; lest the ides of March make me as mad as the March Hare.
NOTE: Is this all that ground-breaking? No. The astute among you see the inherent circular nature of what’s being shown here. Do kids find this interesting? Yes. Is it a doorway for some into more advanced proofs? Yes. I do if for the kids, yo.
How I Teach Calculus: A Comedy (The Chain Rule) Spring Break Reading Assignment
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3 thoughts on “How is a raven like a writing desk? (Quotient Rule -> Product Rule)”
I’m very excited to find this blog! I, too, am constantly on the lookout for better, more student centered methods to present the material and engage students. Thank you for sharing your experiences- I’m taking LOTS of notes!
My mathematical response to that riddle is “There exactly one of each”, although I’m not sure that makes sense to anyone but me.
What I like about this demonstration is that not only do my students get the derivation (pun intended) of the derivative of a quotient, but it also provides a “mnemonic in name only” to remember which term of the numerator is subtracted. This is my primary reason for doing this demonstration.