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:
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.