Answer
$$\frac{dy}{dx} = \sqrt {t^3}e^{-t}(t - \frac{1}{2})$$
Work Step by Step
We will need to use the following formula:
$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$
This formula lets us calculate ${\frac{dy}{dt}}$ and ${\frac{dx}{dt}}$ individually, then take their quotient to obtain $\frac{dy}{dx}$.
$$\frac{dy}{dt} = \frac{d}{dt}(\sqrt t e^{-t}) = \frac{e^{-t}}{2\sqrt t} - \sqrt t e^{-t}$$$$\frac{dx}{dt} = \frac{d}{dt}(\frac{1}{t}) = -\frac{1}{t^2}$$
$$\frac{dy}{dx} = \frac{\frac{e^{-t}}{2\sqrt t} - \sqrt t e^{-t}}{-\frac{1}{t^2}} = \sqrt {t^3}e^{-t}(t - \frac{1}{2})$$
