Transform an exponential series without differentiating again

Let $y=e^{1/(x+1)}$. Show that $yy^{\prime\prime}-(y^\prime)^2=\dfrac{2y^2}{(x+1)^3}$. Hence find its Maclaurin expansion through $x^2$, and deduce the expansion of $e^{-x/(x+1)}$ through $x^2$.