一. 简答题（共1题，100分）

1. (简答题) 设函数\begin{aligned} Z=Z(x,y) \end{aligned}由方程\begin{aligned} x^2y-{\rm e}^{xz}=\sin y \end{aligned}所确定，求\begin{aligned} \frac{\partial z}{\partial x},\frac{\partial z}{\partial y} \end{aligned}.

解：\begin{aligned} F(x,y,z)=x^2-{\rm e}^{xz}-\sin y \end{aligned},\begin{aligned} F_x'(x,y,z)=2xy-z{\rm e}^{xz} \end{aligned}

\begin{aligned} F_y'(x,y,z)=x^2-\cos y \end{aligned},\begin{aligned} F_z'(x,y,z)=-x{\rm e}^{xz} \end{aligned}

\therefore\begin{aligned} \frac{\partial z}{\partial x}=-\frac{F_x'(x,y,z)}{F_z'(x,y,z)}=\frac{2xy-z{\rm e}^{xz}}{x{\rm e}^{xz}} \end{aligned}

\begin{aligned} \frac{\partial z}{\partial y}=-\frac{F_y'(x,y,z)}{F_z'(x,y,z)}=\frac{x^2-\cos y}{x{\rm e}^{xz}} \end{aligned}