# 复合函数的偏导数

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

\begin{array}{l}\displaystyle\frac{\partial z}{\partial x}=\frac{\partial f}{\partial u}\cdot\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\cdot\frac{\partial v}{\partial x}\\ \displaystyle\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}

2，如果函数中即有自变量，也有中间变量，这会比较麻烦一点，如果只有一个自变量

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

3，若函数中即有自变量，也有中间变量，中间变量和自变量都多于一个，

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

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

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

\begin{align*}\frac{\partial z}{\partial t}&=\frac{\partial f}{\partial x}\cdot\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\cdot\frac{\partial y}{\partial t}\\ &=2x\sin y\cdot (2t)+x^2\cos y\cdot 2s\\ &=4tx\sin y+2sx^2\cos y\end{align*}

\begin{align*}\frac{\partial u}{\partial y}&=\frac{\partial f}{\partial y}+\frac{\partial f}{\partial z}\cdot\frac{\partial z}{\partial y}\\ &=x\sin z+xy\cos z\cdot e^{x^2+y}\end{align*}

\begin{align*}\frac{\partial u}{\partial x}&=\frac{\partial f}{\partial s}\cdot\frac{\partial s}{\partial x}+\frac{\partial f}{\partial t}\cdot\frac{\partial t}{\partial x}\\ &=f_1’\cdot 2x+f_2’\cdot ye^{xy}\end{align*}

\begin{align*}\frac{\partial u}{\partial y}&=\frac{\partial f}{\partial s}\cdot\frac{\partial s}{\partial y}+\frac{\partial f}{\partial t}\cdot\frac{\partial t}{\partial y}\\ &=f_1’\cdot (-2y)+f_2’\cdot xe^{xy}\end{align*}