📝 题目
例 4 设 $u = f\left( {x,\frac{x}{y}}\right)$ ,求 $\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}$ .
💡 答案与解析
解 我们不引入中间变量 $s = x$ 和 $t = x/y$ ,用 ${f}_{i}^{\prime }$ 表示对函数 $f$ 第 $i$ 个中间变量求偏导数,这样我们有
$$ \frac{\partial u}{\partial x} = {f}_{1}^{\prime } \cdot 1 + {f}_{2}^{\prime } \cdot \frac{1}{y} = {f}_{1}^{\prime } + \frac{1}{y}{f}_{2}^{\prime }, $$
$$ \frac{\partial u}{\partial y} = {f}_{1}^{\prime } \cdot 0 + {f}_{2}^{\prime } \cdot \left( {-\frac{x}{{y}^{2}}}\right) = - \frac{x}{{y}^{2}}{f}_{2}^{\prime }. $$