复合函数求导法则：
[F(x,y,z)]^\prime = (xyz)^\prime = x^\prime yz+xy^\prime z+xy z^\prime
规则是：分别把变量的导数与除该变量外其余部分相乘，然后把结果相加。
除该变量外其余部分就是该变量的偏导数，可得：
[F(x,y,z)]^\prime = x^\prime F_x+ y^\prime F_y+ z^\prime F_z
对于自变量为x的方程：
F(x,y)=0
等式两边同时对x求导：
x^\prime F_x+ y^\prime F_y = 0
整理可得：
\frac{dy}{dx} = -\frac{F_x}{F_y}
F(x,y,z)=0
x^\prime F_x+ y^\prime F_y+ z^\prime F_z = 0
对于自变量为x,y的二元方程：
分别对x,y求偏导：
F_x+ \frac{dz}{dx} F_z = 0
F_y+ \frac{dz}{dy} F_z = 0
对于方程组
\left\{\begin{array}{l} F(x, y, z)=0 \\ G(x, y, z)=0 \end{array}\right.
先确定谁是自变量，这里x是自变量。 对x求导：
\left\{\begin{array}{l} F_x+F_y \frac{d y}{d x}+F_z \frac{d z}{d x}=0 \\ G_x+G_y \frac{d y}{d x}+G_z \frac{d z}{d x}=0 \end{array}\right.
解方程组即可求出 \frac{d y}{d x},\frac{d z}{d x} 。
\left\{\begin{array}{l} F(x, y, u, v)=0 \\ G(x, y, u, v)=0 \end{array}\right.
先确定谁是自变量，这里x,y是自变量。
对x求导可得：
\left\{\begin{array}{l} F_x+F_u \frac{\partial u}{\partial x}+F_v \frac{\partial v}{\partial x}=0 \\ G_x+G_u \frac{\partial u}{\partial x}+G_v \frac{\partial v}{\partial x}=0 \end{array}\right.
对y求导可得：
\left\{\begin{array}{l} F_y+F_u \frac{\partial u}{\partial y}+F_v \frac{\partial v}{\partial y}=0 \\ G_y+G_u \frac{\partial u}{\partial y}+G_v \frac{\partial v}{\partial y}=0 \end{array}\right.
解方程组即可求出 \frac{\partial u}{\partial x}, \frac{\partial v}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial y}
