Quotient rule for functions is straightforward: $g(x)/h(x)$
Can I find a pair of functions not constant where the quotient of the derivative is the derivative of the quotient?
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what do you mean by "quotient of the derivative"? Do you mean $f'/g'$? – JMJ Apr 25 '17 at 00:20
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Yes! That's what I meant – Lock Apr 25 '17 at 00:26
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Related: https://math.stackexchange.com/q/10927/120540 . There, the question asks in addition for $(f \cdot g)' = f' \cdot g'$ – pjs36 Apr 25 '17 at 00:59
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Suppose neither $f$ nor $g$ vanish. We need a function such that $$ f'/g' = (gf'-fg')/g^2. $$
Thus
$$ g^2f' = gg'f' - f(g')^2 \rightarrow g(g'-g)f' = f(g')^2 \rightarrow f'/f = \frac{(g')^2}{(g'-g)g}, $$ and therefore $$ \ln f = \int\frac{(g')^2}{(g'-g)g}dx. $$
Putting in $g(x) = e^{-x}$ gives $f(x) = e^{x}$ which gives the desired behavior.
JMJ
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