##### 偏导数运算 - 偏导数运算 - **偏导数运算**是对[[偏导数]]进行的[[运算]]和[[运算律]], 主要是与代数运算的可交换性和一些计算定理, 具有[[高阶偏导数]]和[[黑塞矩阵]]等 - 和差 - $\displaystyle\frac{\partial}{\partial x} [f(x, y) \pm g(x, y)] = \frac{\partial f}{\partial x} \pm \frac{\partial g}{\partial x}$ - 乘积 - $\displaystyle \frac{\partial}{\partial x} [f(x, y)g(x, y)] = f(x, y)\frac{\partial g}{\partial x} + g(x, y)\frac{\partial f}{\partial x}$ - 除法 - $\displaystyle \frac{\partial}{\partial x} \left( \frac{f(x, y)}{g(x, y)} \right) = \frac{g(x, y)\frac{\partial f}{\partial x} - f(x, y)\frac{\partial g}{\partial x}}{g(x, y)^2}$ - 链式求导 (多元复合函数求导) - 如果 $z=f(u,v)$, 其中 $u=u(x,y)$ 且 $v=v(x,y)$ - $\displaystyle\frac{\partial z}{\partial x} = \frac{\partial f}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial x}$ - $\displaystyle \frac{\partial z}{\partial y} = \frac{\partial f}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial y}$ - 如果 $z=f(x_1,x_2,\dots,x_m)$, 其中 $x_i=x_i(t_1,t_2,\dots,t_n)$, $i\in{1,\dots,m}$ - $\displaystyle\frac{\partial z}{\partial x_j} = \frac{\partial f}{\partial x_1}\frac{\partial x_1}{\partial t_j} + \frac{\partial f}{\partial x_2}\frac{\partial x_2}{\partial t_j}+\dots + \frac{\partial f}{\partial x_m}\frac{\partial x_m}{\partial t_j}$