Question - Partial Derivative Operations of Logarithmic Function
Solution:
题目要求对函数 $$ z = \ln(xy) $$ 进行偏导运算。首先，找到 $$ \frac{\partial^3 z}{\partial x^2 \partial y} $$ 和 $$ \frac{\partial^3 z}{\partial x \partial y^2} $$，然后计算它们的差值。函数 $$ z $$ 关于 $$ x $$ 的一阶偏导数是：\[ \frac{\partial z}{\partial x} = \frac{1}{xy} \cdot y = \frac{1}{x} \]函数 $$ z $$ 关于 $$ y $$ 的一阶偏导数是：\[ \frac{\partial z}{\partial y} = \frac{1}{xy} \cdot x = \frac{1}{y} \]接下来，计算它们的二阶偏导数和三阶偏导数。对 $$ \frac{\partial z}{\partial x} $$ 关于 $$ x $$ 求二阶偏导数：\[ \frac{\partial^2 z}{\partial x^2} = -\frac{1}{x^2} \]对 $$ \frac{\partial^2 z}{\partial x^2} $$ 关于 $$ y $$ 求三阶偏导数：\[ \frac{\partial^3 z}{\partial x^2 \partial y} = 0 \]对 $$ \frac{\partial z}{\partial y} $$ 关于 $$ y $$ 求二阶偏导数：\[ \frac{\partial^2 z}{\partial y^2} = -\frac{1}{y^2} \]对 $$ \frac{\partial^2 z}{\partial y^2} $$ 关于 $$ x $$ 求三阶偏导数：\[ \frac{\partial^3 z}{\partial x \partial y^2} = 0 \]由于 $$ \frac{\partial^3 z}{\partial x^2 \partial y} $$ 和 $$ \frac{\partial^3 z}{\partial x \partial y^2} $$ 都等于 0，所以：\[ \frac{\partial^3 z}{\partial x^2 \partial y} - \frac{\partial^3 z}{\partial x \partial y^2} = 0 - 0 = 0 \]因此，所求的差值是 0。
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