复合函数的求导法则

Chain Rule

复合函数求导时，要分析函数可看作哪些函数复合而成

核心就是明确变量关系，区分导数 $\frac{d}{d}$ 和偏导数 $\frac{\partial}{\partial}$

$u = g (x)$ 在 $x$ 可导， $y = f (u)$ 在 $u = g (x)$ 可导，则复合函数 $y = f [g (x)]$ 在点 $x$ 处可导，导数为：

\begin{array}{r} \frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x} \end{array}

核心是：全微分形式不变性

\begin{matrix} z = f (u, v) u = φ (t) v = ψ (t) \\ \frac{d z}{d t} = \frac{\partial z}{\partial u} \frac{d u}{d t} + \frac{\partial z}{\partial v} \frac{d v}{d t} \end{matrix}

\begin{matrix} z = f (u, v) u = φ (x, y) v = ψ (x, y) \\ \frac{\partial z}{\partial x} = \frac{\partial z}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial z}{\partial v} \frac{\partial v}{\partial x} \frac{\partial z}{\partial y} = \frac{\partial z}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial z}{\partial v} \frac{\partial v}{\partial y} \end{matrix}

\begin{matrix} z = f (u, v) u = φ (x, y) v = ψ (y) \\ \frac{\partial z}{\partial x} = \frac{\partial z}{\partial u} \frac{\partial u}{\partial x} \frac{\partial z}{\partial y} = \frac{\partial z}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial z}{\partial v} \frac{d v}{d y} \end{matrix}

\begin{matrix} z = f (u, x, y) u = φ (x, y) \\ \frac{\partial z}{\partial x} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial f}{\partial x} \frac{\partial z}{\partial y} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial f}{\partial y} \end{matrix}

引入记号： $f_{1}^{'} (u, v) = f_{u} (u, v) f_{2}^{'} = f_{v} (u, v) f_{12}^{'} = f_{u v} (u, v)$