设由 F(x, y, z) = 0 确定了隐函数 z = f(x, y)，且 (partial F)/(partial x) bigg|_((1,1,1)) = -1，(partial F)/(partial y) bigg|_((1,1,1)) = 2，(partial z)/(partial y) bigg|_((1,1)) = 1，则 (partial z)/(partial x) bigg|_((1,1)) = ( )。A. -(1)/(2)B. (1)/(2)C. -2D. 2

本题考查隐函数求偏导数的知识。解题思路是利用隐函数求偏导数的公式来计算$\frac{\partial z}{\partial x}$的值。

已知由$F(x, y, z) = 0$确定了隐函数$z = f(x, y)$，根据隐函数求偏导数的公式：
$\frac{\partial z}{\partial x}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}}$，$\frac{\partial z}{\partial y}=-\frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial z}}$。

我们可以通过$\frac{\partial z}{\partial y}$的值求出$\frac{\partial F}{\partial z}$的值，再代入$\frac{\partial z}{\partial x}$的公式中计算。

• 步骤一：根据$\frac{\partial z}{\partial y}$的值求出$\frac{\partial F}{\partial z}$的值
  已知$\frac{\partial z}{\partial y}=-\frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial z}}$，且$\frac{\partial F}{\partial y} \bigg|_{(1,1,1)} = 2$，$\frac{\partial z}{\partial y} \bigg|_{(1,1)} = 1$，将其代入公式可得：
  $1=-\frac{2}{\frac{\partial F}{\partial z}}$
  等式两边同时乘以$\frac{\partial F}{\partial z}$可得：
  $\frac{\partial F}{\partial z}=-2$

• 步骤二：将$\frac{\partial F}{\partial x}$和$\frac{\partial F}{\partial z}$的值代入$\frac{\partial z}{\partial x}$的公式中计算
  已知$\frac{\partial z}{\partial x}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial z}}$，且$\frac{\partial F}{\partial x} \bigg|_{(1,1,1)} = -1$，$\frac{\partial F}{\partial z}=-2$，将其代入公式可得：
  $\frac{\partial z}{\partial x}=-\frac{-1}{-2}=-\frac{1}{2}$