§8.5 多元复合函数求导法则（链式法则）

概述

对于抽象复合函数（$f$ 无具体表达式），需要用链式法则。这是期末与考研必考题。关键是画出变量关系图。

抽象函数记号约定：

• $f_1^{\prime }$ 或 $f_x^{\prime }$：对第一个位置变量求偏导
• $f_2^{\prime }$ 或 $f_y^{\prime }$：对第二个位置变量求偏导
• $f_{11}^{\prime \prime }$：对第一个位置变量求两次偏导
• $f_{12}^{\prime \prime }$：先对第一个再对第二个

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§8.5.1 三种复合情形与链式法则

情形一：多元与一元的复合（全导数）

$z=f(u,v)$，$u=\phi (x)$，$v=\psi (x)$，则 $z=f(\phi (x),\psi (x))$ 是 $x$ 的一元函数：

$\frac{dz}{dx}=\frac{\partial z}{\partial u}\cdot \frac{du}{dx}+\frac{\partial z}{\partial v}\cdot \frac{dv}{dx}$

或称全导数。

推论（中间变量多于两个）： $\frac{dz}{dx}=\frac{\partial z}{\partial u}\frac{du}{dx}+\frac{\partial z}{\partial v}\frac{dv}{dx}+\frac{\partial z}{\partial w}\frac{dw}{dx}$

例1：$z=x^2-y^2$，$x=\sin t$，$y=\cos t$，求 $\frac{dz}{dt}$

$\frac{dz}{dt}=2x\cos t-(-2y)(-\sin t)=4\sin t\cos t=2\sin 2t$

例2（抽象函数）：$z=f(\sin t,\cos t)$，$f$ 可微，求 $\frac{dz}{dt}$

$\frac{dz}{dt}=f_1^{\prime }\cdot \cos t+f_2^{\prime }\cdot (-\sin t)=\cos t\cdot f_1^{\prime }(\sin t,\cos t)-\sin t\cdot f_2^{\prime }(\sin t,\cos t)$

特殊情形：$x$ 既是自变量又是中间变量 $z=f(u,x),u=\phi (x)⟹\frac{dz}{dx}=\frac{\partial f}{\partial u}\cdot \frac{du}{dx}+\frac{\partial f}{\partial x}$

例：$z=f(t^2,t)$，$\frac{dz}{dt}=f_1^{\prime }\cdot 2t+f_2^{\prime }$

最简情形：$z=f(x,x)$ → $\frac{dz}{dx}=f_1^{\prime }(x,x)+f_2^{\prime }(x,x)$

情形二：多元与多元的复合

$z=f(u,v)$，$u=\phi (x,y)$，$v=\psi (x,y)$，则：

$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\cdot \frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\cdot \frac{\partial v}{\partial x}$

$\frac{\partial z}{\partial y}=\frac{\partial z}{\partial u}\cdot \frac{\partial u}{\partial y}+\frac{\partial z}{\partial v}\cdot \frac{\partial v}{\partial y}$

例3：$z=f(xy,x+y)$，$f$ 可微，求 $\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}$

$\frac{\partial z}{\partial x}=y{f}_1^{\prime }+f_2^{\prime }$，$\frac{\partial z}{\partial y}=x{f}_1^{\prime }+f_2^{\prime }$

注：当 $x$ 同时是自变量和中间变量时： $z=f(u,x,y)$，$u=\phi (x,y)$ $\frac{\partial z}{\partial x}=\frac{\partial f}{\partial u}\cdot \frac{\partial u}{\partial x}+\frac{\partial f}{\partial x}$

例4：$z=f(y\sin x,x)$，$\frac{\partial z}{\partial x}=f_1^{\prime }\cdot y\cos x+f_2^{\prime }$，$\frac{\partial z}{\partial y}=f_1^{\prime }\cdot \sin x$

情形三：一元与多元的复合

$z=f(u)$，$u=\phi (x,y)$，则：

$\frac{\partial z}{\partial x}=f^{\prime }(u)\cdot \frac{\partial u}{\partial x},\frac{\partial z}{\partial y}=f^{\prime }(u)\cdot \frac{\partial u}{\partial y}$

例5：$z=f(x^2+y^2)$，$f$ 可微，求偏导数

$\frac{\partial z}{\partial x}=f^{\prime }\cdot 2x$，$\frac{\partial z}{\partial y}=f^{\prime }\cdot 2y$

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§8.5.2 复合函数的高阶偏导数（核心难点）

关键提醒

$f_1^{\prime }$（或 $f_u^{\prime }$）仍然是复合函数，与 $f$ 有相同的复合结构**！求二阶导时不能遗漏对 $f_1^{\prime }$ 再使用链式法则。**

典型例题

例6：$z=f(xy,x+y)$，$f$ 具有二阶连续偏导数，求 $\frac{{\partial }^{2}z}{\partial x\partial y}$

$\frac{\partial z}{\partial x}=y{f}_1^{\prime }+f_2^{\prime }$

$\begin{array}{rl}\frac{{\partial }^{2}z}{\partial x\partial y}& =\frac{\partial }{\partial y}(y{f}_1^{\prime }+f_2^{\prime })\\ & =f_1^{\prime }+y(f_{11}^{\prime \prime }\cdot x+f_{12}^{\prime \prime }\cdot 1)+(f_{21}^{\prime \prime }\cdot x+f_{22}^{\prime \prime }\cdot 1)\\ & =f_1^{\prime }+xy{f}_{11}^{\prime \prime }+(y+x)f_{12}^{\prime \prime }+f_{22}^{\prime \prime }\end{array}$

例7：$z=f(xy,\frac{y}{x})$，求二阶偏导 $\frac{{\partial }^{2}z}{\partial x^2},\frac{{\partial }^{2}z}{\partial y^2},\frac{{\partial }^{2}z}{\partial x\partial y}$

$\frac{\partial z}{\partial x}=y{f}_1^{\prime }-\frac{y}{x^2}{f}_2^{\prime }$ $\frac{\partial z}{\partial y}=x{f}_1^{\prime }+\frac{1}{x}{f}_2^{\prime }$

$\frac{{\partial }^{2}z}{\partial x^2}=y^2{f}_{11}^{\prime \prime }-\frac{2y^2}{x^2}{f}_{12}^{\prime \prime }+\frac{y^2}{x^4}{f}_{22}^{\prime \prime }+\frac{2y}{x^3}{f}_2^{\prime }$

例8：$z=x^{3}f(xy,\frac{y}{x})$，求二阶偏导

$\frac{\partial z}{\partial x}=3x^{2}f+x^3(y{f}_1^{\prime }-\frac{y}{x^2}{f}_2^{\prime })=3x^{2}f+x^{3}y{f}_1^{\prime }-xy{f}_2^{\prime }$

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§8.5.3 全微分形式不变性

性质：无论 $u,v$ 是自变量还是中间变量，$z=f(u,v)$ 的全微分形式不变： $dz=\frac{\partial z}{\partial u}du+\frac{\partial z}{\partial v}dv=f_u^{\prime }du+f_v^{\prime }dv$

应用方法：逐层微分

对复合函数，直接从外层向内层逐层微分：

$z=f(u,v)$，$u=\phi (x,y)$，$v=\psi (x,y)$

$dz=f_u^{\prime }du+f_v^{\prime }dv=f_u^{\prime }(u_{x}dx+u_{y}dy)+f_v^{\prime }(v_{x}dx+v_{y}dy)$

然后合并 $dx,dy$ 的系数即得 $\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}$。

例9：$F(x,y,z)=0$ 确定 $z=z(x,y)$，求 $\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}$

$dF=F_{x}dx+F_{y}dy+F_{z}dz=0$，解得 $dz=-\frac{F_x}{F_z}dx-\frac{F_y}{F_z}dy$

故 $\frac{\partial z}{\partial x}=-\frac{F_x}{F_z}$，$\frac{\partial z}{\partial y}=-\frac{F_y}{F_z}$

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四、齐次函数的欧拉定理

若 $f(tx,ty,tz)=t^{k}f(x,y,z)$（$k$ 次齐次函数），则：

$x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}+z\frac{\partial f}{\partial z}=kf$

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关键词汇总

情形	公式
多元→一元（全导数）	$\frac{dz}{dx}=\frac{\partial z}{\partial u}\frac{du}{dx}+\frac{\partial z}{\partial v}\frac{dv}{dx}$
多元→多元	$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial x}$
一元→多元	$\frac{\partial z}{\partial x}=f^{\prime }(u)\frac{\partial u}{\partial x}$
全微分不变性	$dz=f_u^{\prime }du+f_v^{\prime }dv$（无论 $u,v$ 是否为中间变量）
隐函数求导	$F(x,y,z)=0⟹\frac{\partial z}{\partial x}=-\frac{F_x}{F_z}$
欧拉定理	$x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=kf$（$k$ 次齐次）
关键技巧	$f_1^{\prime },f_2^{\prime }$ 仍是复合函数，二阶导需再次链式法则