Example 2 (to $4^{\circ}$). Find $y^{\prime}$ if $e^{y}+x y=e$. Also find $y^{\prime}(0)$.
Solution. Differentiate both sides of the given equality, keeping in mind that $y=y(x): e^{y} y^{\prime}+y+x y^{\prime}=0$, from which we find $y^{\prime}=-\frac{y}{e^{y}+x}$. In particular, if $x=0$, then $y=1$ and $y^{\prime}(0)=-\frac{1}{e}$.