偏导数的定义

（定义）函数$z=f(x,y)$在点$P_{0}(x_{0},y_{0})$关于自变量$x$的偏导数$$\frac{\partial f}{\partial x}|_{(x_{0},y_{0})}=\displaystyle \lim_{\Delta x \to 0}\frac{\Delta _{x}z}{\Delta x}=\displaystyle \lim_{\Delta x \to 0}\frac{f(x_{0}+\Delta x,y_{0})-f(x_{0},y_{0})}{\Delta x}$$
$z=f(x,y)$的偏导函数的记号$$\frac{\partial z}{\partial x},\frac{\partial z}{\partial y};\frac{\partial f}{\partial x},\frac{\partial f}{\partial y};z_{x},z_{y};f_{x}(x,y),f_{y}(x,y);$$$$f'_{1}(x,y),f'_{2}(x,y);f_{x},f_{y};f'_{1},f'_{2}$$
由偏导数的定义可知，求$f_{x}(x,y)$实际上就是把$f(x,y)$中的$y$看作常数，从而把$f(x,y)$当作$x$的一元函数求导。
偏导数的符号是一个整体符号，不能看作分子与分母的商。

偏导数的几何意义

$f_{x}(x_{0},y_{0})$就是曲线$$C_{1}:\left\{\begin{matrix}z=f(x，y) \\y=y_{0}\end{matrix}\right.$$在点$$M_{0}(x_{0},y_{0},f(x_{0},y_{0}))$$处的切线对x轴的斜率。
切向量$T_{x}=(1,0,f(x_{0},y_{0}))$

二阶偏导数共有四种 $$\frac{\partial }{\partial x}\left ( \frac{\partial z}{\partial x} \right )=\frac{\partial ^{2}z}{\partial x^{2}}=\frac{\partial ^{2}f}{\partial x^{2}}=z_{xx}=f_{xx}(x,y)$$ $$\frac{\partial }{\partial y}\left ( \frac{\partial z}{\partial x} \right )=\frac{\partial ^{2}z}{\partial x^{}\partial y^{}}=\frac{\partial ^{2}f}{\partial x^{}\partial y^{}}=z_{xy}=f_{xy}(x,y)$$ $$\frac{\partial }{\partial x}\left ( \frac{\partial z}{\partial y} \right )=\frac{\partial ^{2}z}{\partial y^{}\partial x^{}}=\frac{\partial ^{2}f}{\partial y^{}\partial x^{}}=z_{yx}=f_{yx}(x,y)$$ $$\frac{\partial }{\partial y}\left ( \frac{\partial z}{\partial y} \right )=\frac{\partial ^{2}z}{\partial y^{2}}=\frac{\partial ^{2}f}{\partial y^{2}}=z_{yy}=f_{yy}(x,y)$$ 其中，第二行称为：z先对x后对y的二阶混合偏导数。
（定理）如果函数$f$在区域D内处处存在直到k阶的所有偏导数且所有这些偏导数都在D内连续，则$f$在D内的k阶混合偏导数与求导的次序无关。