<s>[INST] Different results implicit differentiation When trying to differentiate $\frac{x+y}{xy}=x$, I get different results. If I use the quotient rule:
$$
\begin{array}{rcl}
\dfrac{xy(1+y´)-(x+y)(xy´+y)}{(xy)^2} &=& 1 \\
\dfrac{xy+xyy´-x^2y´-xy-xyy´-y^2}{(xy)^2} &=& 1 \\
-x^2y´ -y^2 &=& x^2y^2 \\
y´ &=& \dfrac{-(y^2+x^2y^2)}{x^2}
\end{array}
$$
On the other hand if I do the following
$$
\begin{array}{rcl}
\dfrac{x+y}{xy} &=& x \\
x+y &=& x^2y \\
1+y´ &=& x^2y´ +2xy \\
y´ -x^2y´ &=& 2xy-1 \\
y´ &=& \dfrac{2xy-1}{1-x^2}
\end{array}
$$
I don't understand what's happening.
 [/INST] $\frac{x+y}{xy}=x$ so that $y=\frac{x}{x^2-1}$ If we use this we can show that two terms for $y'$ are equal
</s>