Example Question - quotient calculation

Partial Derivative Quotient Calculation of Ln(xy)

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 \(\frac{x}{y}\)。