题目一：两个变量的三元函数
已知函数 $u$ $=$ $f \left( x + y , x y , \frac { x } { y } \right)$, 求 $\frac{\partial^{2} u}{\partial x^{2} }$, $\frac { \partial^{2} u }{ \partial x \partial y }$, $\frac{ \partial^{2} u }{\partial y^{2}}$.
其中，$f$ 具有二阶连续偏导数。
难度评级：
解析一
为了方便计算，我们首先进行如下标注：
$$
\tag{1}
\begin{cases}
x + y & \leftrightarrow 1 \\
xy & \leftrightarrow 2 \\
\frac{x}{y} & \leftrightarrow 3
\end{cases}
$$
同时，在上面的 $(1)$ 式中，分别对 $x$ 和 $y$ 求偏导数，有：
$$
\tag{2}
\begin{cases}
x + y & \leftrightarrow 1 \begin{cases}
\partial x \rightarrow \textcolor{orangered}{1} \\
\partial y \rightarrow \textcolor{orangered}{1}
\end{cases} \\ \\
xy & \leftrightarrow 2 \begin{cases}
\partial x \rightarrow \textcolor{orangered}{y} \\
\partial y \rightarrow \textcolor{orangered}{x}
\end{cases} \\ \\
\frac{x}{y} & \leftrightarrow 3 \begin{cases}
\partial x \rightarrow \textcolor{orangered}{\frac{1}{y}} \\
\partial y \rightarrow \textcolor{orangered}{\frac{-x}{y^{2}}}
\end{cases}
\end{cases}
$$
接着，先求解一阶偏导数：
$$
\begin{aligned}
\frac{ \partial u }{ \partial x } & = f_{1}^{ \prime } + y f_{2}^{ \prime } + \frac{1}{y} f_{3}^{\prime} \\ \\
\frac{ \partial u }{ \partial y } & = f_{1}^{\prime} + x f_{2}^{ \prime } – \frac{x}{y^{2} } f_{3}^{\prime}
\end{aligned}
$$
于是：
$$
\begin{aligned}
\frac{ \partial^{2} u }{ \partial x^{2} } \\ \\
= & \ \frac{\partial u}{\partial x} \left( f_{1}^{ \prime } + y f_{2}^{ \prime } + \frac{1}{y} f_{3}^{\prime} \right) \\ \\
= & \ f_{11}^{ \prime \prime } + y f_{12}^{ \prime \prime } + \frac{1}{y} f_{13}^{\prime \prime} \\
– & \ y \left( f_{21}^{ \prime \prime } + y f_{22}^{ \prime \prime } + \frac{1}{y} f_{23}^{ \prime \prime } \right) \\
– & \ \frac{1}{y} \left( f_{31}^{ \prime \prime } + y f_{32}^{ \prime \prime } + \frac{1}{y} f_{33}^{ \prime \prime } \right) \\ \\
\Rightarrow & \ \text{ 二阶偏导数连续 } \Rightarrow \textcolor{gray}{\begin{cases}
f ^{\prime \prime}_{12} = f ^{\prime \prime}_{21} \\
f ^{\prime \prime}_{13} = f ^{\prime \prime}_{31} \\
f ^{\prime \prime}_{23} = f ^{\prime \prime}_{32} \\
\end{cases}} \\ \\
= & \ \textcolor{springgreen}{\boldsymbol{ f_{11}^{ \prime \prime } + 2y f_{12}^{ \prime \prime } + \frac{2}{y} f_{13}^{\prime \prime} + y^{2} f_{22}^{\prime \prime} + 2 f_{23}^{\prime \prime} + \frac{1}{y^{2} } f_{33}^{\prime \prime} }}
\end{aligned}
$$
于是：
$$
\begin{aligned}
\frac { \partial^{2} u }{\partial x \partial y} \\ \\
= & \ \frac{\partial u}{\partial y} \left( f_{1}^{ \prime } + y f_{2}^{ \prime } + \frac{1}{y} f_{3}^{\prime} \right) \\ \\
= & \ f_{11}^{\prime \prime} + x f_{12}^{\prime \prime} – \frac{x}{y^{2} } f_{13}^{\prime \prime} + f_{2}^{\prime} \\
– & \ y \left( f_{21}^{\prime \prime} + x f_{22}^{\prime \prime} – \frac{x}{y^{2} } f_{23}^{\prime \prime} \right) – \frac{1}{y^{2} } f_{3}^{\prime} \\
– & \ \frac{1}{y} \left( f_{31}^{ \prime \prime } + x f_{32}^{\prime \prime} – \frac {x}{y^{2}} f_{33}^{\prime \prime} \right) \\ \\
= & \ \textcolor{springgreen}{\boldsymbol{ f_{11}^{\prime \prime} + (x+y ) f_{12}^{\prime \prime} + \left( \frac{1}{y} – \frac{x}{y^{2} } \right) f_{13}^{\prime \prime} }} \\
& \textcolor{springgreen}{\boldsymbol{ + f_{2}^{\prime} + x y f_{22}^{\prime \prime} – \frac{1}{y^{2} } f_{3}^{\prime} – \frac{x}{y^{3} } f_{33}^{\prime \prime} }}
\end{aligned}
$$
于是：
$$
\begin{aligned}
\frac { \partial^{2} u }{\partial y^{2} } \\ \\
= & \ \frac{\partial u}{\partial y} \left( f_{1}^{ \prime } + y f_{2}^{ \prime } + \frac{1}{y} f_{3}^{\prime} \right) \\ \\
= & \ f_{11}^{\prime \prime} + x f_{12}^ {\prime \prime} – \frac{x}{y^{2}} f_{13}^{\prime \prime} \\
– & \ x \left(f_{21}^{\prime \prime} + x f_{22}^{\prime \prime} – \frac{x}{y^{2}} f_{23}^{\prime \prime} \right) + \frac{2x}{y^{3}} f_{3}^{\prime} \\
– & \ \frac{x}{y^{2} } \left( f_{31}^{\prime \prime} + x f_{32}^{\prime \prime} – \frac{x}{y^{2} } f_{33}^{\prime \prime} \right) \\ \\
= & \ \textcolor{springgreen}{\boldsymbol{ f_{11}^{\prime \prime} + 2 x f_{12}^{\prime \prime} – 2 \frac{x}{y^{2} } f_{13}^{\prime \prime} + x^{2} f_{22}^{\prime \prime} }} \\
& \textcolor{springgreen}{\boldsymbol{- 2 \frac { x^{2} }{y^{2}} f_{23}^{\prime \prime} + \frac{ x^{2} }{y^{4} } f_{33}^{\prime \prime} + \frac{2x}{y^{3} } f_{3}^{\prime} }}
\end{aligned}
$$