设函数 $z=z(x, y)$ 由 $e^z+x z=2 x-y$ 确定, 则 $\left.\frac{\partial^2 z}{\partial^2 x}\right|_{(1,1)}=$
【答案】 $-\frac{3}{2}$

【解析】 两边同时对 $\mathrm{x}$ 求导得: $\mathrm{e}^z \cdot \frac{\partial z}{\partial x}+z+x \cdot \frac{\partial z}{\partial x}=2-0$ (1)
两边再同时对 $x$ 求导得: $\mathrm{e}^2 \cdot \frac{\partial z}{\partial x} \cdot \frac{\partial z}{\partial x}+e^2 \cdot \frac{\partial^2 z}{\partial x^2}+\frac{\partial z}{\partial x}+\frac{\partial z}{\partial x}+x \cdot \frac{\partial^2 z}{\partial x^2}=0$
将 $x=1, y=1$ 代入原方程得 $e^x+z=1 \Rightarrow z=0$
$$
\begin{aligned}
& \text { 代入(1)式得 } e^0 \cdot \frac{\partial z}{\partial x}+0+\frac{\partial z}{\partial x}=2 \Rightarrow \frac{\partial z}{\partial x}=1 \\
& \text { 代入(2)式得 } e^0 \cdot 1+e^0 \cdot \frac{\partial^2 z}{\partial x^2}+1+1+\frac{\partial^2 z}{\partial x^2}=0 \Rightarrow \frac{\partial^2 z}{\partial x^2}=-\frac{3}{2} \text {. }
\end{aligned}
$$
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