已知 z = f(x, xy, x^2 + y^2) 具有连续偏导数，则下面正确的是(）。 A. (partial z)/(partial x) = f_1'； B. (partial z)/(partial x) = f_1' + yf_2' + 2xf_3'； C. (partial z)/(partial x) = f_1' + f_2' + f_3'； D. (partial z)/(partial x) = f_1' + xf_2' + 2yf_3'。

设 $ u = x $，$ v = xy $，$ w = x^2 + y^2 $，则 $ z = f(u, v, w) $。由链式法则得： $\frac{\partial z}{\partial x} = f_1' \cdot \frac{\partial u}{\partial x} + f_2' \cdot \frac{\partial v}{\partial x} + f_3' \cdot \frac{\partial w}{\partial x}$ 计算各偏导数： $\frac{\partial u}{\partial x} = 1, \quad \frac{\partial v}{\partial x} = y, \quad \frac{\partial w}{\partial x} = 2x$ 代入得： $\frac{\partial z}{\partial x} = f_1' \cdot 1 + f_2' \cdot y + f_3' \cdot 2x = f_1' + yf_2' + 2xf_3'$ 答案：B