1. 偏导数的定义

设函数\(z = f(x,y)\)在点\((x_{0},y_{0})\)的某一邻域内有定义，当\(y = y_{0}\)固定不变，而\(x\)在\(x_{0}\)处有增量\(\Delta x\)时，相应地函数有增量\(f(x_{0}+\Delta x,y_{0})-f(x_{0},y_{0})\)，如果极限\(\lim\limits_{\Delta x\rightarrow0}\frac{f(x_{0}+\Delta x,y_{0})-f(x_{0},y_{0})}{\Delta x}\)存在，则称此极限为函数\(z = f(x,y)\)在点\((x_{0},y_{0})\)处对\(x\)的偏导数，记作\(f_{x}(x_{0},y_{0})\)，\(\frac{\partial z}{\partial x}\vert_{(x_{0},y_{0})}\)，\(z_{x}\vert_{(x_{0},y_{0})}\)或\(\frac{\partial f}{\partial x}\vert_{(x_{0},y_{0})}\)。

同理，当\(x = x_{0}\)固定不变，\(y\)在\(y_{0}\)处有增量\(\Delta y\)时，若极限\(\lim\limits_{\Delta y\rightarrow0}\frac{f(x_{0},y_{0}+\Delta y)-f(x_{0},y_{0})}{\Delta y}\)存在，则称此极限为函数\(z = f(x,y)\)在点\((x_{0},y_{0})\)处对\(y\)的偏导数，记作\(f_{y}(x_{0},y_{0})\)，\(\frac{\partial z}{\partial y}\vert_{(x_{0},y_{0})}\)，\(z_{y}\vert_{(x_{0},y_{0})}\)或\(\frac{\partial f}{\partial y}\vert_{(x_{0},y_{0})}\)。

例如，对于函数\(z = x^{2}+3y\)，求\(z\)在点\((1,2)\)处对\(x\)的偏导数，把\(y = 2\)看作常数，对\(x\)求导得\(z_{x}=2x\)，将\(x = 1\)代入得\(z_{x}\vert_{(1,2)} = 2\)。

2. 偏导数的计算方法

直接求导法：对于多元函数，当求关于某一个变量的偏导数时，将其他变量看作常数，然后按照一元函数的求导法则进行求导。

例如，对于函数\(u = x^{2}y + y^{2}z + z^{2}x\)，求\(\frac{\partial u}{\partial x}\)，把\(y\)和\(z\)看作常数，对\(x\)求导得\(\frac{\partial u}{\partial x}=2xy + z^{2}\)。

复合函数偏导数法则（链式法则）：设\(z = f(u,v)\)，\(u = u(x,y)\)，\(v = v(x,y)\)，则\(\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\cdot\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\cdot\frac{\partial v}{\partial x}\)，\(\frac{\partial z}{\partial y}=\frac{\partial z}{\partial u}\cdot\frac{\partial u}{\partial y}+\frac{\partial z}{\partial v}\cdot\frac{\partial v}{\partial y}\)。

例如，设\(z = u^{2}+v\)，\(u = xy\)，\(v = x + y\)，求\(\frac{\partial z}{\partial x}\)。

首先，\(\frac{\partial z}{\partial u}=2u\)，\(\frac{\partial z}{\partial v}=1\)，\(\frac{\partial u}{\partial x}=y\)，\(\frac{\partial v}{\partial x}=1\)。

根据链式法则\(\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\cdot\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\cdot\frac{\partial v}{\partial x}=2u\cdot y+1\cdot1 = 2xy\cdot y + 1=2xy^{2}+1\)。

3. 偏导数的几何意义

对于二元函数\(z = f(x,y)\)，\(f_{x}(x_{0},y_{0})\)表示曲面\(z = f(x,y)\)与平面\(y = y_{0}\)的交线在点\((x_{0},y_{0},z_{0})\)（其中\(z_{0}=f(x_{0},y_{0})\)）处的切线对\(x\)轴的斜率；\(f_{y}(x_{0},y_{0})\)表示曲面\(z = f(x,y)\)与平面\(x = x_{0}\)的交线在点\((x_{0},y_{0},z_{0})\)处的切线对\(y\)轴的斜率。

4. 高阶偏导数

设函数\(z = f(x,y)\)在区域\(D\)内具有偏导数\(\frac{\partial z}{\partial x}=f_{x}(x,y)\)和\(\frac{\partial z}{\partial y}=f_{y}(x,y)\)，如果这两个偏导数又存在偏导数，则称它们是函数\(z = f(x,y)\)的二阶偏导数。

二阶偏导数有四个：\(\frac{\partial^{2}z}{\partial x^{2}}=f_{xx}(x,y)=\left(\frac{\partial z}{\partial x}\right)_{x}\)，\(\frac{\partial^{2}z}{\partial y^{2}}=f_{yy}(x,y)=\left(\frac{\partial z}{\partial y}\right)_{y}\)，\(\frac{\partial^{2}z}{\partial x\partial y}=f_{xy}(x,y)=\left(\frac{\partial z}{\partial x}\right)_{y}\)，\(\frac{\partial^{2}z}{\partial y\partial x}=f_{yx}(x,y)=\left(\frac{\partial z}{\partial y}\right)_{x}\)。

例如，对于函数\(z = x^{3}y^{2}\)，\(\frac{\partial z}{\partial x}=3x^{2}y^{2}\)，\(\frac{\partial z}{\partial y}=2x^{3}y\)。

二阶偏导数：\(\frac{\partial^{2}z}{\partial x^{2}} = 6xy^{2}\)，\(\frac{\partial^{2}z}{\partial y^{2}}=2x^{3}\)，\(\frac{\partial^{2}z}{\partial x\partial y}=6x^{2}y\)，\(\frac{\partial^{2}z}{\partial y\partial x}=6x^{2}y\)。

在一定条件下（函数的二阶偏导数连续），\(f_{xy}(x,y)=f_{yx}(x,y)\)。

高等数学