Hello everyone, first time poster here. I've been a lurker for about a week, but finally decided to join because I cannot for the life of me figure out this problem. 1. The problem statement, all variables and given/known data Zach has trouble with the Quotient Rule; he thinks that d/dx (f(x)/g(x)) = f′(x)/g′(x). On his last calculus test, Zach applied this erroneous rule to a quotient in which g(x) = x. Somehow, he managed to get the right answer. What are all the possibilities for the function f(x)? 2. Relevant equations d/dx (f(x)/g(x)) = f′(x)/g′(x) (Zach's rule) d/dx (f(x)/g(x)) = [f′(x)g(x) - f(x)g′(x)] / (g(x))^2 (actual quotient rule) 3. The attempt at a solution I first attempted to set the derivative using Zach's equation equal to the derivative using the quotient rule. After simplification, I got f(x) = (x-x^2)f′(x) I am not really sure what to do at this point however. I am assuming it is something either really obvious, or really not-so-obvious. I tried using a substitution, and even an hour of trial and error, but I am just not seeing it. Am I going to have to use the limit definition of the derivative? Any help would be greatly appreciated.