例. 设二元隐函数 $u=u(x,y)$, $v=v(x,y)$ 由方程组
\[
\begin{cases}
2x=v^2-u^2,\\
y=uv,
\end{cases}
\]
确定, 求 $\dfrac{\partial u}{\partial x}$, $\dfrac{\partial u}{\partial y}$, $\dfrac{\partial v}{\partial x}$, $\dfrac{\partial v}{\partial y}$.
1
方程组两边分别对 $x$ 求导(求偏导数), 得
\[
\begin{cases}
2=2v\frac{\partial v}{\partial x}-2u\frac{\partial u}{\partial x},\\
0=\frac{\partial u}{\partial x}v+u\frac{\partial v}{\partial x}.
\end{cases}
\]
即
\[
\begin{pmatrix}
-2u & 2v\\
v & u
\end{pmatrix}
\begin{pmatrix}
\frac{\partial u}{\partial x}\\
\frac{\partial v}{\partial x}
\end{pmatrix}=
\begin{pmatrix}
2\\
0
\end{pmatrix}
\]