I know that
$\dfrac{d(xy)}{dx} = y$
but what does
$\dfrac{dx}{d(xy)} =\, ?$
I know this is an odd equation, but it comes from some ugly change of variables and I am stuck with it.
$\endgroup$ 14 Answers
$\begingroup$Yet another answer: $ \dfrac{du}{dv} \dfrac{dv}{du} = 1$ by chain rule. This means the answer is $\dfrac{1}{y}$.
$\endgroup$ 3 $\begingroup$In order to conclude that $\frac{d(xy)}{dx}$, one needs to know that $\frac{dy}{dx}=0$, which would make $y$ a constant. I note this because typically $y$ refers to a variable that might change with $x$. In this case,the product rule tells us that $$\frac{d(xy)}{dx}=y\frac{dx}{dx}+x\frac{dy}{dx}=$$ $$y+x\frac{dy}{dx}$$. Then $$\frac{dx}{d(xy)}=\frac{1}{y+x\frac{dy}{dx}}.$$ Under the assumption that $\frac{dy}{dx}=0$, or alternatively that $y$ is a constant, the other answers are correct. I did want to give a warning that $y$ typically is not a constant however.
$\endgroup$ $\begingroup$Let $z=xy$. Then $d(x)/d(xy) = d(xy y^{-1})/d(xy) = d(zy^{-1})/d(z)$.
$\endgroup$ $\begingroup$Or, in an easier way, if x and y are independent (and you know that because d(xy)/dx=y), you can treat y as a constant. so the answer is: 1/y
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