Example 1. Find $d^{2} z$, if $z=\frac{x+y}{x-y}$.
Solution. Sequential differentiation of the given function yields:

$$
\begin{gathered}
z_{x}^{\prime}=\frac{x-y-x-y}{(x-y)^{2}}=-\frac{2 y}{(x-y)^{2}}, \quad z_{y}^{\prime}=\frac{x-y+x+y}{(x-y)^{2}}=\frac{2 x}{(x-y)^{2}} \\
z_{x^{2}}^{\prime \prime}=\frac{4 y}{(x-y)^{3}}, \quad z_{x y}^{\prime \prime}=-2 \frac{x+y}{(x-y)^{3}}, \quad z_{y^{2}}^{\prime \prime}=\frac{4 x}{(x-y)^{2}}
\end{gathered}
$$

Therefore,

$d^{2} z=\frac{4 y}{(x-y)^{3}} d x^{2}-4 \frac{x+y}{(x-y)^{3}} d x d y+4 \frac{x}{(x-y)^{3}} d y^{2}=4 \frac{y d x^{2}-(x+y) d x d y+z d y^{2}}{(x-y)^{3}}$.