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 $\frac{\mathrm{d}}{\mathrm{d}}$ 和偏导数 $\frac{\partial }{\partial }$

一、一元复合函数的求导法则（链式法则）

$u=g(x)$ 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：

\dfrac{\mathrm{d} y}{\mathrm{d} x} =\dfrac{\mathrm{d} y}{\mathrm{d} u} \cdot \dfrac{\mathrm{d} u}{\mathrm{d} x} \end{align}$$ ### 二、多元复合函数的求导法则 核心是：全微分形式不变性 #### 1. 一元函数与多元函数复合 $$\begin{gathered} z =f(u,v)\quad u=\varphi(t)\quad v=\psi(t) \\ \\ \dfrac{\mathrm{d} z}{\mathrm{d} t} =\dfrac{\partial z}{\partial u}\dfrac{\mathrm{d} u}{\mathrm{d} t}+\dfrac{\partial z}{\partial v}\dfrac{\mathrm{d} v}{\mathrm{d} t} \end{gathered}$$ #### 2. 多元函数与多元函数复合 $$\begin{gathered} z =f(u,v)\quad u=\varphi(x,y)\quad v=\psi(x,y) \\ \\ \dfrac{\partial z}{\partial x} =\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x}+\dfrac{\partial z}{\partial v}\dfrac{\partial v}{\partial x} \quad \quad \dfrac{\partial z}{\partial y} =\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial y}+\dfrac{\partial z}{\partial v}\dfrac{\partial v}{\partial y} \end{gathered}$$ #### 3. 混合情形 $$\begin{gathered} z =f(u,v)\quad u=\varphi(x,y)\quad v=\psi(y) \\ \\ \dfrac{\partial z}{\partial x} =\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x} \quad \quad \dfrac{\partial z}{\partial y} =\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial y}+\dfrac{\partial z}{\partial v}\dfrac{\mathrm{d} v}{\mathrm{d} y} \end{gathered}$$ #### 4. 中间变量本身是复合函数的自变量 $$\begin{gathered} z =f(u,x,y)\quad u=\varphi(x,y) \\ \\ \dfrac{\partial z}{\partial x}=\dfrac{\partial f}{\partial u}\dfrac{\partial u}{\partial x}+\dfrac{\partial f}{\partial x}\quad \dfrac{\partial z}{\partial y} =\dfrac{\partial f}{\partial u}\dfrac{\partial u}{\partial y}+\dfrac{\partial f}{\partial y} \end{gathered}$$ 引入记号：$f_{1}'(u,v)=f_{u}(u,v)\quad f_{2}'=f_{v}(u,v)\quad f_{12}'=f_{uv}(u,v)$