kaoyan2advanced 高等数学 第209题

**答案**：$\displaystyle \frac{\partial^{2} z}{\partial x \partial y} = -2 f''(2x-y) + g_{uu} \cdot x + g_{uv} \cdot x^{2} + g_{v} + g_{vu} \cdot y + g_{vv} \cdot xy$ **解析**： 步骤1：令 $u=2x-y$，$v=x$，$w=xy$，则 $z=f(u)+g(v,w)$。$\displaystyle \frac{\partial z}{\partial x} = f'(u) \cdot 2 + g_v \cdot 1 + g_w \cdot y = 2f' + g_v + y g_w$。 步骤2：$\displaystyle \frac{\partial^{2} z}{\partial x \partial y} = \frac{\partial}{\partial y}(2f' + g_v + y g_w) = 2f'' \cdot (-1) + (g_{vv} \cdot 0 + g_{vw} \cdot x) + (1 \cdot g_w + y (g_{wv} \cdot 0 + g_{ww} \cdot x)) = -2f'' + x g_{vw} + g_w + xy g_{ww}$。注意 $g_{vw}=g_{uv}$，$g_{ww}=g_{vv}$，故答案为 $-2f''(2x-y) + x g_{uv}(x,xy) + g_v(x,xy) + xy g_{vv}(x,xy)$。 **难度**：★★★☆☆