Question

QC: You might be wondering: why do the product and quotient rules look so

different from each other? An article I read gave more similar-looking forms:

(f*g)' = (f*g)*(f'/f + g'/g)

(f/g)' = (f/g)*(f'/f - g'/g)

(College Math Journal 42:4 September 2011 page 323, by Roger Eggleton and Vladimir Kustov; it also gives rules for 3 functions at a time, and second derivatives)

i) Do a little algebra to show that these indeed are equivalent to the rules we learned in the book.

ii) What is the mathematical downside to using the product rule in this new form (not including “easier to make a mistake” or “more complicated”)?

iii) Let's think of a useful, natural interpretation for f'/f or g'/g: if f(5)=100 and f'(5)=3, then f is increasing at 3% per unit time, right? How is this related to f'/f ?

Answer 1