二阶偏导数的定义与符号
本节提供了关于如何计算二阶偏导数的详细解答示例。
对于一个二元函数 \( f(x, y) \)，我们可以定义四个二阶偏导数及其标准符号：
\[ \begin{aligned} \dfrac{\partial^2 f}{\partial x^2} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial x} \right) = \dfrac{\partial}{\partial x} (f_x) = (f_x)_x = f_{xx} \\\\ \dfrac{\partial^2 f}{\partial y^2} &= \dfrac{\partial}{\partial y} \left( \dfrac{\partial f}{\partial y} \right) = \dfrac{\partial}{\partial y} (f_y) = (f_y)_y = f_{yy} \\\\ \dfrac{\partial^2 f}{\partial y \partial x} &= \dfrac{\partial}{\partial y} \left( \dfrac{\partial f}{\partial x} \right) = \dfrac{\partial}{\partial y} (f_x) = (f_x)_y = f_{xy} \\\\ \dfrac{\partial^2 f}{\partial x \partial y} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial y} \right) = \dfrac{\partial}{\partial x} (f_y) = (f_y)_x = f_{yx} \end{aligned} \]二阶偏导数详细解答例题
点击每个例题以查看其详细的分步解答。
例题 1
已知 \( f(x, y) = \sin(xy) \)，求 \( f_{xx} \) 和 \( f_{yy} \)。
显示例题 1 的解答
计算 \( f_{xx} \)：
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}\sin(xy)\right) = \frac{\partial}{\partial x}(y \cos(xy)) = -y^2 \sin(xy) \]计算 \( f_{yy} \)：
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}\sin(xy)\right) = \frac{\partial}{\partial y}(x \cos(xy)) = -x^2 \sin(xy) \]例题 2
已知 \( f(x, y) = x^3 + 2xy \)，求 \( f_{xx} \)、\( f_{yy} \)、\( f_{xy} \) 和 \( f_{yx} \)。
显示例题 2 的解答
计算 \( f_{xx} \)：
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3 + 2xy)\right) = \frac{\partial}{\partial x}(3x^2 + 2y) = 6x \]计算 \( f_{yy} \)：
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3 + 2xy)\right) = \frac{\partial}{\partial y}(2x) = 0 \]计算 \( f_{xy} \)：
\[ f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3 + 2xy)\right) = \frac{\partial}{\partial y}(3x^2 + 2y) = 2 \]计算 \( f_{yx} \)：
\[ f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3 + 2xy)\right) = \frac{\partial}{\partial x}(2x) = 2 \]例题 3
已知 \( f(x, y) = x^3y^4 + x^2y \)，求 \( f_{xx} \)、\( f_{yy} \)、\( f_{xy} \) 和 \( f_{yx} \)。
显示例题 3 的解答
计算 \( f_{xx} \)：
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial x}(3x^2y^4 + 2xy) = 6xy^4 + 2y \]计算 \( f_{yy} \)：
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial y}(4x^3y^3 + x^2) = 12x^3y^2 \]计算 \( f_{xy} \)：
\[ f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial y}(3x^2y^4 + 2xy) = 12x^2y^3 + 2x \]计算 \( f_{yx} \)：
\[ f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial x}(4x^3y^3 + x^2) = 12x^2y^3 + 2x \]