u(x,y)=\frac{\frac{\partial z}{\partial x} - \frac{\partial z}{\partial y}}{x - y} (1)z = x^2 + y^2 - \varphi(x + y + z)(2)求：\frac{\partial u}{\partial x} = ?解：\frac{\partial z}{\partial x} = 2x - \varphi'(\frac{\partial z}{\partial x} + 1) (3)\frac{\partial z}{\partial y} = 2y - \varphi'(\frac{\partial z}{\partial y} + 1)(4)由(3)、(4)分別解出：\frac{\partial z}{\partial x} = \frac{2x - \varphi'}{1 + \varphi'} (5)\frac{\partial z}{\partial y} = \frac{2y - \varphi'}{1 + \varphi'} (6)將(5)、(6)代入(1)式，得到：u(x,y) = \frac{\frac{\partial z}{\partial x} - \frac{\partial z}{\partial y}}{x - y}= \frac{2}{1 + \varphi'}(7)這就是第二問題的第一步。而\frac{\partial u}{\partial x} = -\frac{2\varphi''(1 + \frac{\partial z}{\partial x})}{(1 + \varphi')^2}將（5）式代入，最后得到：\frac{\partial u}{\partial x} = -\frac{2\varphi''(1 + 2x)}{(1 + \varphi')^3} (8)這是第二問題的最后一步！