Find $\displaystyle \frac {\delta y}{\delta x} \, \left ( e^{-y} \right )$,
$\displaystyle = e^{-y} \cdot - \frac {\delta y}{\delta x}$
$\displaystyle = -e^{-y} \cdot \frac {\delta y}{\delta x}$, is this correct?
I don't know what conventions you have been taught, but to me your
notation doesn't mean anything.
It looks as though you are asked to find:
$\displaystyle \frac{d}{dx} e^{-y}$,
for which:
$\displaystyle \frac{d}{dx} e^{-y} = \frac{dy}{dx}\left(\frac{d}{dy}e^{-y}\right)=-e^{-y}\, \frac{dy}{dx}$.
RonL