1. **State the problem:** Given the equation $xy=1$, and the rate of change $\frac{dx}{dt}=12$ when $x=2$ and $y=\frac{1}{2}$, find $\frac{dy}{dt}$.\n\n2. **Use implicit differentiation:** Differentiate both sides of $xy=1$ with respect to $t$. Using the product rule, we get:\n$$\frac{d}{dt}(xy) = \frac{d}{dt}(1)$$\n$$x\frac{dy}{dt} + y\frac{dx}{dt} = 0$$\n\n3. **Solve for $\frac{dy}{dt}$:** Rearranging the equation:\n$$x\frac{dy}{dt} = -y\frac{dx}{dt}$$\n$$\frac{dy}{dt} = \frac{-y\frac{dx}{dt}}{x}$$\n\n4. **Substitute the known values:** $x=2$, $y=\frac{1}{2}$, and $\frac{dx}{dt}=12$:\n$$\frac{dy}{dt} = \frac{-\frac{1}{2} \times 12}{2}$$\n\n5. **Simplify the expression:**\n$$\frac{dy}{dt} = \frac{-6}{2}$$\n$$\frac{dy}{dt} = -3$$\n\n**Final answer:** $\frac{dy}{dt} = -3$