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In his Cameos for Calculus (page 12) Roger B. Nelsen gives the following "proof" of the quotient rule as an application of the product rule: $$ w=\frac{u}{v}\implies u=vw $$ then $$ u'=(vw)'=v'w+vw' $$ therefore solving for $w'$ yields $$ w'=\frac{u'v-v'u}{v^{2}}. $$ and then he mentions that

"But there is a serious flaw in this "proof." What is it?"

The proof looks fine to me and I'm curious to know what Prof. Nelson might have had in mind as a serious flaw in this argument. Any comments are appreciated.

Simon
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This proof assumes that we already know that the quotient $w=\frac{u}{v}$ is differentiable, while part of the point of the quotient rule is to actually prove this fact.

carmichael561
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  • This might have been the point of the author, but I wouldn't call it a serious flaw because I can interpret the rule differently (or in a weaker form): If $w=u/v$ is the quotient of two differentiable functions then $w'=\frac{u'v-vu'}{v^2}$ at any point of differentiability of $w$. – Simon Mar 01 '16 at 03:14
  • @Simon: You are right, but then we have a problem in using the Rule, since in principle we cannot know that the answer we get is correct, since the point of interest might not be a point of differentiability. – André Nicolas Mar 01 '16 at 03:25