Question

Suppose xy=2 and dy/dt=1. Find dx/dt when x=3

Answer 1

If dy / dt = 1, then y = t. If y = t, then y = 2/3, and then you could do 3 /2/3, getting us 3/2

Answer 2

$\bf xy=2\implies y=\cfrac{2}{x}\\\\ -----------------------------\\\\ \cfrac{dx}{dt}\cdot y\quad +\quad x\cdot \cfrac{dy}{dt}=0\implies \cfrac{dx}{dt}=-\cfrac{x}{y}\cdot \cfrac{dy}{dt} \\\\\\ \textit{now, we know that } \begin{cases} \frac{dy}{dt}=1\\ x=3\\ y=\frac{2}{x} \end{cases}\qquad thus \\\\\\ \cfrac{dx}{dt}=-\cfrac{3}{\frac{2}{3}}\cdot 1\implies \cfrac{dx}{dt}=-\cfrac{9}{2}$