$y f^{\prime \prime}(x y)+\varphi^{\prime}(x+y)+y \varphi^{\prime \prime}(x+y)$

#### 解析：

\begin{aligned} \frac{\partial z}{\partial x}=& \frac{\partial}{\partial x}\left[\frac{1}{x} f(x y)+y \varphi(x+y)\right]=-\frac{1}{x^{2}} f(x y)+\frac{y}{x} f^{\prime}(x y)+y \varphi^{\prime}(x+y) \\ \frac{\partial^{2} z}{\partial x \partial y} &=\frac{\partial}{\partial y}\left(-\frac{1}{x^{2}} f(x y)+\frac{y}{x} f^{\prime}(x y)+y \varphi^{\prime}(x+y)\right) \\ &=-\frac{1}{x^{2}} f^{\prime}(x y) x+\frac{1}{x} f^{\prime}(x y)+\frac{y}{x} f^{\prime \prime}(x y) x+\varphi^{\prime}(x+y)+y \varphi^{\prime \prime}(x+y) \\ &=-\frac{1}{x} f^{\prime}(x y)+\frac{1}{x} f^{\prime}(x y)+y f^{\prime \prime}(x y)+\varphi^{\prime}(x+y)+y \varphi^{\prime \prime}(x+y) \\ &=y f^{\prime \prime}(x y)+\varphi^{\prime}(x+y)+y \varphi^{\prime \prime}(x+y) \end{aligned}

\begin{aligned} \frac{\partial z}{\partial y} &=\frac{\partial}{\partial y}\left[\frac{1}{x} f(x y)+y \varphi(x+y)\right]=\frac{1}{x} f^{\prime}(x y) x+\varphi(x+y)+y \varphi^{\prime}(x+y) \\ &=f^{\prime}(x y)+\varphi(x+y)+y \varphi^{\prime}(x+y) \\ \frac{\partial^{2} z}{\partial x \partial y} &=\frac{\partial^{2} z}{\partial y \partial x}=\frac{\partial}{\partial x}\left[f^{\prime}(x y)+\varphi(x+y)+y \varphi^{\prime}(x+y)\right] \\ &=y f^{\prime \prime}(x y)+\varphi^{\prime}(x+y)+y \varphi^{\prime \prime}(x+y) \end{aligned}

\begin{aligned} \frac{\partial^{2} z}{\partial x \partial y} &=\frac{\partial}{\partial x}\left[\frac{\partial}{\partial y}\left(\frac{1}{x} f(x y)\right)\right]+\frac{\partial}{\partial y}\left[\frac{\partial}{\partial x}(y \varphi(x+y))\right] \\ &=\frac{\partial}{\partial x}\left[\frac{1}{x} f^{\prime}(x y) x\right]+\frac{\partial}{\partial y}\left[y \varphi^{\prime}(x+y)\right] \\ &=\frac{\partial}{\partial x}\left[f^{\prime}(x y)\right]+\frac{\partial}{\partial y}\left[y \varphi^{\prime}(x+y)\right] \\ &=y f^{\prime \prime}(x y)+\varphi^{\prime}(x+y)+y \varphi^{\prime \prime}(x+y) \end{aligned}