# 二阶及更高阶偏导数

## 例子及其解答

### 例1解答与详细步骤

1. 一阶偏导数
a. 对 $$x$$ 求导： $\dfrac{\partial f}{\partial x} = \dfrac{\partial}{\partial x}(x^2y) + \dfrac{\partial}{\partial x}(3xy^2) \\\\ = 2xy + 3y^2$ b. 对 $$y$$ 求导： $\dfrac{\partial f}{\partial y} = \dfrac{\partial}{\partial y}(x^2y) + \dfrac{\partial}{\partial y}(3xy^2) \\\\ = x^2 + 6xy$ 2. 二阶偏导数
a. 对 $$\dfrac{\partial f}{\partial x}$$ 再对 $$x$$ 求导： $\dfrac{\partial^2 f}{\partial x^2} = \dfrac{\partial}{\partial x} \left(\dfrac{\partial f}{\partial x} \right) \\\\ =\dfrac{\partial}{\partial x}(2xy + 3y^2) = 2y$ b. 对 $$\dfrac{\partial f}{\partial y}$$ 再对 $$y$$ 求导： $\dfrac{\partial^2 f}{\partial y^2} = \dfrac{\partial}{\partial y} \left(\dfrac{\partial f}{\partial y} \right) \\\\ = \dfrac{\partial}{\partial y}(x^2 + 6xy) = 6x$ c. 对 $$\dfrac{\partial f}{\partial x}$$ 对 $$y$$ 求导： $\dfrac{\partial^2 f}{\partial x \partial y} = \dfrac{\partial }{\partial y} \left(\dfrac{\partial f}{\partial x}\right) \\\\= \dfrac{\partial}{\partial y}(2xy + 3y^2) = 2x + 6y$ d. 对 $$\dfrac{\partial f}{\partial y}$$ 对 $$x$$ 求导（由于对称性，如果函数的混合偏导数是连续的，结果应与步骤3相同）： $\dfrac{\partial^2 f}{\partial y \partial x} = \dfrac{\partial }{\partial x} \left(\dfrac{\partial f}{\partial y}\right) \\\\ =\dfrac{\partial}{\partial x}(x^2 + 6xy) = 2x + 6y$ 注意 $$\dfrac{\partial^2 f}{\partial y \partial x} = \dfrac{\partial^2 f}{\partial x \partial y}$$，这说明了在一定连续性条件下克莱劳定理关于混合偏导数相等的结论。

### 例2解答与详细步骤

1. 一阶偏导数
a. 对 $$x$$ 求导： $\dfrac{\partial g}{\partial x} = \dfrac{\partial}{\partial x}(e^{xy}) + \dfrac{\partial}{\partial x}(\sin(x)y^2) \\\\= ye^{xy} + \cos(x)y^2$ b. 对 $$y$$ 求导： $\dfrac{\partial g}{\partial y} = \dfrac{\partial}{\partial y}(e^{xy}) + \dfrac{\partial}{\partial y}(\sin(x)y^2) \\\\= xe^{xy} + 2y\sin(x)$ 2. 二阶偏导数
a. 对 $$\dfrac{\partial g}{\partial x}$$ 再对 $$x$$ 求导： $\dfrac{\partial^2 g}{\partial x^2} = \dfrac{\partial}{\partial x} \left(\dfrac{\partial g}{\partial x}\right) \\\\=\dfrac{\partial}{\partial x}(ye^{xy} + \cos(x)y^2) \\\\= y^2e^{xy} - y^2\sin(x)$ 此步骤涉及对每项分别应用微分的乘积法则。
b. 对 $$\dfrac{\partial g}{\partial y}$$ 再对 $$y$$ 求导： $\dfrac{\partial^2 g}{\partial y^2} = \dfrac{\partial}{\partial y} \left(\dfrac{\partial g}{\partial y}\right) \\\\ = \dfrac{\partial}{\partial y}(xe^{xy} + 2y\sin(x)) \\\\= x^2e^{xy} + 2\sin(x)$ 同样，应用乘积法则，此次重点关注指数项和正弦项如何随 $$y$$ 的变化而变化。
c. 对 $$\dfrac{\partial g}{\partial x}$$ 对 $$y$$ 求导： $\dfrac{\partial^2 g}{\partial x \partial y} = \dfrac{\partial }{\partial y} \left(\dfrac{\partial g}{\partial x}\right) \\\\ = \dfrac{\partial}{\partial y}(ye^{xy} + \cos(x)y^2) \\\\= e^{xy} + xye^{xy} + 2y\cos(x)$ d. 对 $$\dfrac{\partial g}{\partial y}$$ 对 $$x$$ 求导（由于混合偏导数的连续性和对称性，结果与步骤3相同）： $\dfrac{\partial^2 g}{\partial y \partial x} = \dfrac{\partial }{\partial x} \left(\dfrac{\partial g}{\partial y}\right) \\\\ = \dfrac{\partial}{\partial x}(xe^{xy} + 2y\sin(x)) \\\\= e^{xy} + xye^{xy} + 2y\cos(x)$ $$\dfrac{\partial^2 g}{\partial y \partial x} = \dfrac{\partial^2 g}{\partial x \partial y}$$ 的等式说明了在一定连续性条件下克莱劳定理关于混合偏导数相等的结论。

### 例3解答与详细步骤

1. 一阶偏导数
a. 对 $$x$$ 求导： $\dfrac{\partial f}{\partial x} = \dfrac{\partial}{\partial x}(x^3y^2) + \dfrac{\partial}{\partial x}(x^2e^y) \\\\= 3x^2y^2 + 2xe^y$ b. 对 $$y$$ 求导： $\dfrac{\partial f}{\partial y} = \dfrac{\partial}{\partial y}(x^3y^2) + \dfrac{\partial}{\partial y}(x^2e^y) \\\\= 2x^3y + x^2e^y$ 2. 二阶偏导数
a. 对 $$x$$ 两次求导： $\dfrac{\partial^2 f}{\partial x^2} = \dfrac{\partial}{\partial x} \left(\dfrac{\partial f}{\partial x}\right) \\\\ = \dfrac{\partial}{\partial x}(3x^2y^2 + 2xe^y) \\\\= 6xy^2 + 2e^y$ b. 对 $$y$$ 两次求导： $\dfrac{\partial^2 f}{\partial y^2} = \dfrac{\partial}{\partial y} \left(\dfrac{\partial f}{\partial y}\right) \\\\ = \dfrac{\partial}{\partial y}(2x^3y + x^2e^y) \\\\= 2x^3 + x^2e^y$ c. 先对 $$x$$ 后对 $$y$$ 求导： $\dfrac{\partial^2 f}{\partial y \partial x} = \dfrac{\partial}{\partial y}\left(\dfrac{\partial f}{\partial x}\right) \\\\ = \dfrac{\partial}{\partial y}(3x^2y^2 + 2xe^y) \\\\ = 6x^2y + 2xe^y$ d. 先对 $$y$$ 后对 $$x$$ 求导： $\dfrac{\partial^2 f}{\partial x \partial y} = \dfrac{\partial}{\partial x}\left(\dfrac{\partial f}{\partial y}\right)\\\\ = \dfrac{\partial}{\partial x}(2x^3y + x^2e^y) \\\\= 6x^2y + 2xe^y$ 在此例中，我们同样有 $$\dfrac{\partial^2 f}{\partial y \partial x} = \dfrac{\partial^2 f}{\partial x \partial y}$$ 的等式，这说明了在一定连续性条件下克莱劳定理关于混合偏导数相等的结论。
3. 三阶偏导数
a. 先对 $$x$$ 两次，再对 $$y$$ 求导： $\dfrac{\partial^3 f}{\partial y \partial x^2} = \dfrac{\partial}{\partial y} \left(\dfrac{\partial^2 f}{\partial x^2} \right) \\\\ = \dfrac{\partial}{\partial y}(6xy^2 + 2e^y) = 12 x y + 2 e^y$ b. 先对 $$x$$，再对 $$y$$，最后对 $$x$$ 求导： $\dfrac{\partial^3 f}{\partial x \partial y \partial x} = \dfrac{\partial}{\partial x} \left(\dfrac{\partial^2 f}{\partial x \partial y} \right) \\\\ = \dfrac{\partial}{\partial x}(6x^2y + 2xe^y) = 12xy + 2e^y$

## 克莱劳定理

1. 求 $$\dfrac{{\partial f}}{{\partial x}}$$： $\dfrac{{\partial f}}{{\partial x}} = 2xy$ 2. 求 $$\dfrac{{\partial f}}{{\partial y}}$$： $\dfrac{{\partial f}}{{\partial y}} = x^2 + 3y^2$ 现在，我们求混合偏导数：
1. $$\dfrac{{\partial^2 f}}{{\partial x \partial y}}$$： $\dfrac{{\partial^2 f}}{{\partial x \partial y}} = \dfrac{{\partial}}{{\partial x}} \left( \dfrac{{\partial f}}{{\partial y}} \right) = \dfrac{{\partial}}{{\partial x}} (x^2 + 3y^2) = 2x$ 2. $$\dfrac{{\partial^2 f}}{{\partial y \partial x}}$$： $\dfrac{{\partial^2 f}}{{\partial y \partial x}} = \dfrac{{\partial}}{{\partial y}} \left( \dfrac{{\partial f}}{{\partial x}} \right) = \dfrac{{\partial}}{{\partial y}} (2xy) = 2x$ 根据克莱劳定理，由于混合偏导数是连续且相等的，我们有 $$\dfrac{{\partial^2 f}}{{\partial x \partial y}} = \dfrac{{\partial^2 f}}{{\partial y \partial x}}$$。