Question - Partial Derivative Quotient Calculation of Ln(xy)
Solution:
好的，题目要求我们求函数$$ z = \ln(xy) $$的偏导数的商 $$\frac{\partial^3 z}{\partial x^2 \partial y} \div \frac{\partial^3 z}{\partial x \partial y^2}$$ 的值。首先，我们计算 $$ z = \ln(xy) $$ 的一阶和二阶偏导数：一阶偏导数：$$\frac{\partial z}{\partial x} = \frac{1}{xy} \cdot y = \frac{1}{x}$$$$\frac{\partial z}{\partial y} = \frac{1}{xy} \cdot x = \frac{1}{y}$$二阶偏导数：$$\frac{\partial^2 z}{\partial x^2} = -\frac{1}{x^2}$$$$\frac{\partial^2 z}{\partial y^2} = -\frac{1}{y^2}$$$$\frac{\partial^2 z}{\partial x \partial y} = -\frac{1}{xy^2}$$三阶偏导数：$$\frac{\partial^3 z}{\partial x^2 \partial y} = \frac{2}{xy^3}$$$$\frac{\partial^3 z}{\partial x \partial y^2} = \frac{2}{x^2 y^2}$$现在我们可以计算商的值了：$$\frac{\frac{\partial^3 z}{\partial x^2 \partial y}}{\frac{\partial^3 z}{\partial x \partial y^2}} = \frac{\frac{2}{xy^3}}{\frac{2}{x^2 y^2}} = \frac{2}{xy^3} \cdot \frac{x^2 y^2}{2} = \frac{x}{y}$$所以，该导数的商就是 $$\frac{x}{y}$$。
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