3.设 =dfrac (1)(x)f(xy)+yvarphi (x+y), f,φ具有二阶连续导数,则 dfrac ({a)^2z}(axpartial y)= __

步骤 1：计算 $\dfrac{\partial z}{\partial y}$
首先，我们对 $z$ 关于 $y$ 求偏导数。根据链式法则，我们有：
$$
\dfrac{\partial z}{\partial y} = \dfrac{\partial}{\partial y}\left(\dfrac{1}{x}f(xy)\right) + \dfrac{\partial}{\partial y}\left(y\varphi(x+y)\right)
$$
$$
= \dfrac{1}{x}f'(xy)\cdot x + \varphi(x+y) + y\varphi'(x+y)
$$
$$
= f'(xy) + \varphi(x+y) + y\varphi'(x+y)
$$

步骤 2：计算 $\dfrac{\partial^2 z}{\partial x \partial y}$
接下来，我们对 $\dfrac{\partial z}{\partial y}$ 关于 $x$ 求偏导数。根据链式法则，我们有：
$$
\dfrac{\partial^2 z}{\partial x \partial y} = \dfrac{\partial}{\partial x}\left(f'(xy) + \varphi(x+y) + y\varphi'(x+y)\right)
$$
$$
= f''(xy)\cdot y + \varphi'(x+y) + y\varphi''(x+y)
$$
$$
= yf''(xy) + \varphi'(x+y) + y\varphi''(x+y)
$$