Answer
$6\text{ cm}~~\text{per}~~\text{s}$.
Work Step by Step
Using the chain rule it follows:
$$\frac{dy}{dt}=\frac{dy}{dx} \cdot \frac{dx}{dt}$$
Since the $y$ coordinate is decreasing it follows that the rate $\frac{dy}{dt} \lt 0$ therefore:
$$ \frac{dy}{dt}=-3$$
$$-3=\frac{dy}{dx} \cdot \frac{dx}{dt}$$
Using the implicit differentiation to $xy=8$ it follows:
$$(x)'y+x(y)'=(8)'$$
$$y+x\frac{dy}{dx}=0$$
$$\frac{dy}{dx}=-\frac{y}{x}$$
$$\frac{dy}{dx}|_{x=4,y=2}=-\frac{2}{4}=-\frac{1}{2}$$
So:
$$-3=-\frac{1}{2} \cdot \frac{dx}{dt}$$
$$-3\cdot 2 =-\frac{dx}{dt}$$
$$6=\frac{dx}{dt}$$
The answer is $6\text{ cm}~~\text{per}~~\text{s}$.
