24. (本题6分)设 =f(x+y,xy), f具有一阶连续偏导数,求 dfrac (partial z)(partial x), dfrac (partial z)(partial y)

步骤 1：定义中间变量
设 $u = x + y$ 和 $v = xy$，则 $z = f(u, v)$。
步骤 2：应用链式法则求 $\dfrac {\partial z}{\partial x}$
根据链式法则，$\dfrac {\partial z}{\partial x} = \dfrac {\partial f}{\partial u} \cdot \dfrac {\partial u}{\partial x} + \dfrac {\partial f}{\partial v} \cdot \dfrac {\partial v}{\partial x}$。
步骤 3：计算 $\dfrac {\partial u}{\partial x}$ 和 $\dfrac {\partial v}{\partial x}$
$\dfrac {\partial u}{\partial x} = 1$，$\dfrac {\partial v}{\partial x} = y$。
步骤 4：代入计算 $\dfrac {\partial z}{\partial x}$
$\dfrac {\partial z}{\partial x} = \dfrac {\partial f}{\partial u} \cdot 1 + \dfrac {\partial f}{\partial v} \cdot y = \dfrac {\partial f}{\partial u} + \dfrac {\partial f}{\partial v} \cdot y$。
步骤 5：应用链式法则求 $\dfrac {\partial z}{\partial y}$
根据链式法则，$\dfrac {\partial z}{\partial y} = \dfrac {\partial f}{\partial u} \cdot \dfrac {\partial u}{\partial y} + \dfrac {\partial f}{\partial v} \cdot \dfrac {\partial v}{\partial y}$。
步骤 6：计算 $\dfrac {\partial u}{\partial y}$ 和 $\dfrac {\partial v}{\partial y}$
$\dfrac {\partial u}{\partial y} = 1$，$\dfrac {\partial v}{\partial y} = x$。
步骤 7：代入计算 $\dfrac {\partial z}{\partial y}$
$\dfrac {\partial z}{\partial y} = \dfrac {\partial f}{\partial u} \cdot 1 + \dfrac {\partial f}{\partial v} \cdot x = \dfrac {\partial f}{\partial u} + \dfrac {\partial f}{\partial v} \cdot x$。