问题
设函数 $z$ $=$ $f(x, u, v)$, $u$ $=$ $\varphi(x, y)$, $v$ $=$ $\psi(x, y)$, 则 $\frac{\partial z}{\partial x}$ $=$ $?$, $\frac{\partial z}{\partial y}$ $=$ $?$选项
[A]. $\left\{\begin{array}{l} \frac{\partial z}{\partial x}=\frac{\mathrm{d} f}{\mathrm{d} x}+\frac{\mathrm{d} f}{\mathrm{d} u} \cdot \frac{\partial u}{\partial x}+\frac{\mathrm{d} f}{\mathrm{d} v} \cdot \frac{\partial v}{\partial x} \\ \frac{\partial z}{\partial y}=\frac{\mathrm{d} f}{\mathrm{d} u} \cdot \frac{\partial u}{\partial y}+\frac{\mathrm{d} f}{\mathrm{d} v} \cdot \frac{\partial v}{\partial y} \end{array}\right.$[B]. $\left\{\begin{array}{l} \frac{\partial z}{\partial x}=\frac{\partial f}{\partial x} \cdot \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x} \cdot \frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial x} \\ \frac{\partial z}{\partial y}=\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y} \cdot \frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial y} \end{array}\right.$
[C]. $\left\{\begin{array}{l} \frac{\partial z}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x} \cdot \frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial x} \\ \frac{\partial z}{\partial y}=\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y} \cdot \frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial y} \end{array}\right.$
[D]. $\left\{\begin{array}{l} \frac{\partial z}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}+\frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial x} \\ \frac{\partial z}{\partial y}=\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y}+\frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial y} \end{array}\right.$
$\left\{\begin{array}{l} \frac{\partial z}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}+\frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial x} \\ \frac{\partial z}{\partial y}=\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial y}+\frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial y} \end{array}\right.$