Why most people think that :$(fg)'=f' \cdot g'$?

let $f$ and $g$ be two real valued function , I have asked many students what is the derivative of $(fg)'$ they answered me :it is $f' \cdot g'$, then I seek why most people (students) guess that ?

• – quid Apr 14 '18 at 21:33
• I think it is perhaps the most reasonable guess, even though it is wrong. The instinct for "homomorphisms" is not a bad thing! Who in the world has intuition for "derivations"? :) – paul garrett Apr 14 '18 at 22:44
• Also related: mathoverflow.net/questions/181422/… However, I don't really see this error often enough to put into the "most people" category. – Adam Apr 15 '18 at 0:56
• Possible duplicate of Whence the "everything is linear" phenomenon, and what can we do about it? – Git Gud Apr 15 '18 at 16:11
• I've wondered for a while if issues like this are because teaching gives lots of explicit emphasis on "distribute in this situation", but little or no explicit emphasis on "do not distribute in this situation". So without enough counterexamples for balance, students' intuitions are trained to have an overly positive expectation on whether distributing is appropriate. – Hurkyl Apr 17 '18 at 16:49

2 Answers

As mentioned, a probable cause is an implicit reasoning as if every operation were a homomorphism (similar to the implicit reasoning by linearity). Similar errors include $\ln(x+y) = \ln(x)+\ln(y)$, $e^{xy}=e^x e^y$, $\int f(x)g(x) dx = \int f(x) dx \int g(x) dx$, etc.

Such reasoning by (very loose) analogy can be caused by not understanding that whenever some similar rules do hold, they do for a reason. In particular, it is important to actually explain why $e^{x+y}=e^x e^y$ (at least when $x,y\in\mathbb{N}$), why $(f+g)'=f'+g'$, etc. So the answer to give to the error you mention implies discussing other formulas.

One thing that one can also do is to show how erroneous formulas lead to obvious wrongs. For example, if it where true that $(fg)'=f'g'$, then taking $f=1$ would lead to $g'=0$ whatever $g$ actually is.

A. Maybe because the dash is in the exponent place.

B. It's not a bad guess, if you had little info and had to make a quick guess.

C. Humans are not computers and reason by analogy, first.

On the practical side, who cares? Will knowing why people guess that make the problem go away that you need to train them on the power rule? That there is no quick fix to this sort of insight (some quick key that unlocks a puzzle and means kids don't need to do examples and homework and analogies to other functions (e.g. log or sin) where linearity does not hold)?

• I strongly disagree with your last paragraph. Erroneous conceptions are important to understand, as it does help remedy them. Otherwise, you will simply enforce a rule that is not understood, and will have only solved the issue locally, without progress on similar occasion to make such mistakes. – Benoît Kloeckner Apr 15 '18 at 9:16
• It MIGHT help remedy them. And it MIGHT be that the basic methods to rememdy them are the same. I think there is much too much of a thinking here on this site that the key explanation like a key to a lock will improve learning. However, it generally takes many keys (as different people respond to different things). And more often than explanation, what is required is drill, which develops a familiarity (rather than a top down understanding). Bottom line, if I told you "X" spot in the brain, I don't think it would improve the learning outcome for students. – guest Apr 15 '18 at 13:04
• And don't be so "strong" about your beliefs. Be more curious and reflective instead. Pedagogy and teaching methods are not completely understood yet. Your gut intuitions can be wrong. Therefore, being "strong" about something not well understood, is hubris. – guest Apr 15 '18 at 13:06
• It seems to me that the comments above are saying that, since "pedagogy and teaching methods are not completely understood yet," therefore, one should avoid strong views on (e.g.) understanding mistakes. I don't think these will ever be "completely understood." (I'm not sure what that would even mean/entail...) But, for the immediate context here: the importance of error recognition has been long recognized in the field of mathematics education. And so I share the strong disagreement articulated by @BenoîtKloeckner. – Benjamin Dickman Apr 15 '18 at 16:34
• You don't have to completely understand something to have a hypothesis. But you should be wary of having too strong of a hypothesis. Especially on a topic that is complex and poorly understood. And in general, I find high end math teachers here too confident about their understanding of pedagogy and psychology, perhaps because of their strong understanding of math...not realizing that these are more complex topics (multifactorial, statistical) that they have not reflected on enough. – guest Apr 15 '18 at 21:31