隐函数求偏导专项模拟试卷

隐函数求偏导专项模拟试卷考试时间：120分钟 总分：100分 年级/班级：高三/理科班

隐函数求偏导专项模拟试卷

一、选择题

1.设方程\(z=f(x,y)\)由方程\(F(x,y,z)=0\)确定，其中\(F\)具有连续偏导数，且\(F_x\neq0\)，则\(\frac{\partialz}{\partialx}\)等于

A.\(-\frac{F_y}{F_x}\)

B.\(\frac{F_y}{F_x}\)

C.\(-\frac{F_x}{F_y}\)

D.\(\frac{F_x}{F_y}\)

2.若\(z=z(x,y)\)由方程\(x^2+y^2+z^2=1\)确定，则\(\frac{\partialz}{\partialx}\)在点\((0,0,1)\)处的值为

A.0

B.1

C.-1

D.2

3.设\(z=z(x,y)\)由方程\(e^z=x^2+y^2\)确定，则\(\frac{\partialz}{\partialy}\)等于

A.\(\frac{2y}{e^z}\)

B.\(-\frac{2y}{e^z}\)

C.\(\frac{e^z}{2y}\)

D.\(-\frac{e^z}{2y}\)

4.若\(z=z(x,y)\)由方程\(xyz=1\)确定，则\(\frac{\partialz}{\partialx}\)等于

A.\(-\frac{yz}{x^2}\)

B.\(\frac{yz}{x^2}\)

C.\(-\frac{xz}{y^2}\)

D.\(\frac{xz}{y^2}\)

5.设\(z=z(x,y)\)由方程\(\sin(z)=x+y\)确定，则\(\frac{\partialz}{\partialy}\)等于

A.\(\cos(z)\)

B.\(\frac{1}{\cos(z)}\)

C.\(-\cos(z)\)

D.\(-\frac{1}{\cos(z)}\)

6.若\(z=z(x,y)\)由方程\(x^2+y^2+z^3=3\)确定，则\(\frac{\partialz}{\partialy}\)在点\((1,1,1)\)处的值为

A.0

B.1

C.-1

D.2

7.设\(z=z(x,y)\)由方程\(\ln(z)=x+y\)确定，则\(\frac{\partialz}{\partialx}\)等于

A.\(z\)

B.\(\frac{z}{x}\)

C.\(\frac{z}{y}\)

D.\(z\ln(z)\)

8.若\(z=z(x,y)\)由方程\(x^2+y^2+z^2=2x+2y+2z\)确定，则\(\frac{\partialz}{\partialx}\)等于

A.\(1-\frac{x}{z}\)

B.\(1+\frac{x}{z}\)

C.\(-1+\frac{x}{z}\)

D.\(-1-\frac{x}{z}\)

9.设\(z=z(x,y)\)由方程\(e^{x+y}=z^2\)确定，则\(\frac{\partialz}{\partialy}\)等于

A.\(\frac{z}{e^{x+y}}\)

B.\(-\frac{z}{e^{x+y}}\)

C.\(\frac{z^2}{e^{x+y}}\)

D.\(-\frac{z^2}{e^{x+y}}\)

10.若\(z=z(x,y)\)由方程\(x^3+y^3+z^3=3xyz\)确定，则\(\frac{\partialz}{\partialx}\)等于

A.\(\frac{3xy-x^3}{3xz-y^3}\)

B.\(\frac{3xy+x^3}{3xz+y^3}\)

C.\(\frac{3xy-x^3}{3xz+y^3}\)

D.\(\frac{3xy+x^3}{3xz-y^3}\)

二、填空题

1.设\(z=z(x,y)\)由方程\(x^2+y^2+z^2=1\)确定，则\(\frac{\partial^2z}{\partialx\partialy}\)等于

2.若\(z=z(x,y)\)由方程\(e^z=x^2+y^2\)确定，则\(\frac{\partial^2z}{\partialy^2}\)等于

3.设\(z=z(x,y)\)由方程\(xyz=1\)确定，则\(\frac{\partial^2z}{\partialx\partialy}\)等于

4.若\(z=z(x,y)\)由方程\(\sin(z)=x+y\)确定，则\(\frac{\partial^2z}{\partialx^2}\)等于

5.设\(z=z(x,y)\)由方程\(x^2+y^2+z^3=3\)确定，则\(\frac{\partial^2z}{\partialy\partialx}\)等于

6.若\(z=z(x,y)\)由方程\(\ln(z)=x+y\)确定，则\(\frac{\partial^2z}{\partialx\partialy}\)等于

7.设\(z=z(x,y)\)由方程\(x^2+y^2+z^2=2x+2y+2z\)确定，则\(\frac{\partial^2z}{\partialx\partialy}\)等于

8.若\(z=z(x,y)\)由方程\(e^{x+y}=z^2\)确定，则\(\frac{\partial^2z}{\partialy^2}\)等于

9.设\(z=z(x,y)\)由方程\(x^3+y^3+z^3=3xyz\)确定，则\(\frac{\partial^2z}{\partialx\partialy}\)等于

10.若\(z=z(x,y)\)由方程\(x^2+y^2+z^3=1\)确定，则\(\frac{\partial^2z}{\partialx\partialy}\)等于

三、多选题

1.设\(z=z(x,y)\)由方程\(x^2+y^2+z^2=1\)确定，则下列说法正确的有

A.\(\frac{\partialz}{\partialx}=-\frac{x}{z}\)

B.\(\frac{\partialz}{\partialy}=-\frac{y}{z}\)

C.\(\frac{\partial^2z}{\partialx\partialy}=0\)

D.\(\frac{\partial^2z}{\partialx^2}=\frac{z^2-x^2}{z^3}\)

2.若\(z=z(x,y)\)由方程\(e^z=x^2+y^2\)确定，则下列说法正确的有

A.\(\frac{\partialz}{\partialx}=\frac{2x}{e^z}\)

B.\(\frac{\partialz}{\partialy}=\frac{2y}{e^z}\)

C.\(\frac{\partial^2z}{\partialx\partialy}=0\)

D.\(\frac{\partial^2z}{\partialx^2}=\frac{4x^2-e^z}{e^{2z}}\)

3.设\(z=z(x,y)\)由方程\(xyz=1\)确定，则下列说法正确的有

A.\(\frac{\partialz}{\partialx}=-\frac{yz}{x^2}\)

B.\(\frac{\partialz}{\partialy}=-\frac{xz}{y^2}\)

C.\(\frac{\partial^2z}{\partialx\partialy}=\frac{2xyz}{x^2y^2}\)

D.\(\frac{\partial^2z}{\partialx^2}=\frac{2yz}{x^3}\)

4.若\(z=z(x,y)\)由方程\(\sin(z)=x+y\)确定，则下列说法正确的有

A.\(\frac{\partialz}{\partialx}=\cos(z)\)

B.\(\frac{\partialz}{\partialy}=\cos(z)\)

C.\(\frac{\partial^2z}{\partialx\partialy}=0\)

D.\(\frac{\partial^2z}{\partialx^2}=-\sin(z)\cos(z)\)

5.设\(z=z(x,y)\)由方程\(x^2+y^2+z^3=3\)确定，则下列说法正确的有

A.\(\frac{\partialz}{\partialx}=-\frac{2x}{3z^2}\)

B.\(\frac{\partialz}{\partialy}=-\frac{2y}{3z^2}\)

C.\(\frac{\partial^2z}{\partialx\partialy}=0\)

D.\(\frac{\partial^2z}{\partialx^2}=\frac{2}{3z^2}-\frac{4x^2}{3z^4}\)

6.若\(z=z(x,y)\)由方程\(\ln(z)=x+y\)确定，则下列说法正确的有

A.\(\frac{\partialz}{\partialx}=\frac{z}{x}\)

B.\(\frac{\partialz}{\partialy}=\frac{z}{y}\)

C.\(\frac{\partial^2z}{\partialx\partialy}=0\)

D.\(\frac{\partial^2z}{\partialx^2}=-\frac{z}{x^2}\)

7.设\(z=z(x,y)\)由方程\(x^2+y^2+z^2=2x+2y+2z\)确定，则下列说法正确的有

A.\(\frac{\partialz}{\partialx}=1-\frac{x}{z}\)

B.\(\frac{\partialz}{\partialy}=1-\frac{y}{z}\)

C.\(\frac{\partial^2z}{\partialx\partialy}=0\)

D.\(\frac{\partial^2z}{\partialx^2}=\frac{z-x}{z^2}\)

8.若\(z=z(x,y)\)由方程\(e^{x+y}=z^2\)确定，则下列说法正确的有

A.\(\frac{\partialz}{\partialx}=\frac{z}{e^{x+y}}\)

B.\(\frac{\partialz}{\partialy}=\frac{z}{e^{x+y}}\)

C.\(\frac{\partial^2z}{\partialx\partialy}=0\)

D.\(\frac{\partial^2z}{\partialx^2}=-\frac{z}{e^{x+y}}\)

9.设\(z=z(x,y)\)由方程\(x^3+y^3+z^3=3xyz\)确定，则下列说法正确的有

A.\(\frac{\partialz}{\partialx}=\frac{3xy-x^3}{3xz-y^3}\)

B.\(\frac{\partialz}{\partialy}=\frac{3xy-y^3}{3xz-x^3}\)

C.\(\frac{\partial^2z}{\partialx\partialy}=0\)

D.\(\frac{\partial^2z}{\partialx^2}=\frac{6xy-2x^3-2y^3}{(3xz-y^3)^2}\)

10.若\(z=z(x,y)\)由方程\(x^2+y^2+z^3=1\)确定，则下列说法正确的有

A.\(\frac{\partialz}{\partialx}=-\frac{2x}{3z^2}\)

B.\(\frac{\partialz}{\partialy}=-\frac{2y}{3z^2}\)

C.\(\frac{\partial^2z}{\partialx\partialy}=0\)

D.\(\frac{\partial^2z}{\partialx^2}=\frac{2}{3z^2}-\frac{4x^2}{3z^4}\)

四、判断题

1.若\(z=z(x,y)\)由方程\(x^2+y^2+z^2=1\)确定，则\(\frac{\partial^2z}{\partialx\partialy}=0\)。

2.设\(z=z(x,y)\)由方程\(e^z=x^2+y^2\)确定，则\(\frac{\partialz}{\partialx}=\frac{2x}{e^z}\)。

3.若\(z=z(x,y)\)由方程\(xyz=1\)确定，则\(\frac{\partial^2z}{\partialx\partialy}=\frac{2xyz}{x^2y^2}\)。

4.设\(z=z(x,y)\)由方程\(\sin(z)=x+y\)确定，则\(\frac{\partialz}{\partialy}=\cos(z)\)。

5.若\(z=z(x,y)\)由方程\(x^2+y^2+z^3=3\)确定，则\(\frac{\partial^2z}{\partialy\partialx}=-\frac{\partial^2z}{\partialx\partialy}\)。

6.设\(z=z(x,y)\)由方程\(\ln(z)=x+y\)确定，则\(\frac{\partial^2z}{\partialx\partialy}=0\)。

7.若\(z=z(x,y)\)由方程\(x^2+y^2+z^2=2x+2y+2z\)确定，则\(\frac{\partialz}{\partialx}=1\)。

8.设\(z=z(x,y)\)由方程\(e^{x+y}=z^2\)确定，则\(\frac{\partial^2z}{\partialy^2}=-\frac{z}{e^{x+y}}\)。

9.若\(z=z(x,y)\)由方程\(x^3+y^3+z^3=3xyz\)确定，则\(\frac{\partialz}{\partialx}\)与\(\frac{\partialz}{\partialy}\)的表达式相同。

10.设\(z=z(x,y)\)由方程\(x^2+y^2+z^3=1\)确定，则\(\frac{\partial^2z}{\partialx^2}=-\frac{2x}{z^2}\)。

五、问答题

1.设\(z=z(x,y)\)由方程\(x^2+y^2+z^3=3\)确定，求\(\frac{\partial^2z}{\partialx\partialy}\)。

2.若\(z=z(x,y)\)由方程\(e^z=x^2+y^2\)确定，求\(\frac{\partial^2z}{\partialy^2}\)。

3.设\(z=z(x,y)\)由方程\(xyz=1\)确定，求\(\frac{\partial^2z}{\partialx\partialy}\)及\(\frac{\partial^3z}{\partialx\partialy\partialx}\)。

试卷答案

一、选择题

1.B

解析：由隐函数求导法则，对\(F(x,y,z)=0\)两边关于\(x\)求偏导，得\(F_x+F_z\frac{\partialz}{\partialx}=0\)，解得\(\frac{\partialz}{\partialx}=-\frac{F_x}{F_z}\)。

2.C

解析：对方程\(x^2+y^2+z^2=1\)两边关于\(x\)求偏导，得\(2x+2z\frac{\partialz}{\partialx}=0\)，在点\((0,0,1)\)处，代入得\(\frac{\partialz}{\partialx}=-1\)。

3.A

解析：对方程\(e^z=x^2+y^2\)两边关于\(y\)求偏导，得\(e^z\frac{\partialz}{\partialy}=2y\)，解得\(\frac{\partialz}{\partialy}=\frac{2y}{e^z}\)。

4.A

解析：对方程\(xyz=1\)两边关于\(x\)求偏导，得\(yz+xyz_x=0\)，解得\(\frac{\partialz}{\partialx}=-\frac{yz}{x^2}\)。

5.A

解析：对方程\(\sin(z)=x+y\)两边关于\(y\)求偏导，得\(\cos(z)\frac{\partialz}{\partialy}=1\)，解得\(\frac{\partialz}{\partialy}=\cos(z)\)。

6.C

解析：对方程\(x^2+y^2+z^3=3\)两边关于\(y\)求偏导，得\(2y+3z^2\frac{\partialz}{\partialy}=0\)，在点\((1,1,1)\)处，代入得\(\frac{\partialz}{\partialy}=-1\)。

7.A

解析：对方程\(\ln(z)=x+y\)两边关于\(x\)求偏导，得\(\frac{1}{z}\frac{\partialz}{\partialx}=1\)，解得\(\frac{\partialz}{\partialx}=z\)。

8.D

解析：对方程\(x^2+y^2+z^2=2x+2y+2z\)两边关于\(x\)求偏导，得\(2x+2z\frac{\partialz}{\partialx}=2+2\frac{\partialz}{\partialx}\)，解得\(\frac{\partialz}{\partialx}=\frac{2-x}{z-1}\)。

9.A

解析：对方程\(e^{x+y}=z^2\)两边关于\(y\)求偏导，得\(e^{x+y}\frac{\partialz}{\partialy}=2z\frac{\partialz}{\partialy}\)，解得\(\frac{\partialz}{\partialy}=\frac{z}{e^{x+y}}\)。

10.A

解析：对方程\(x^3+y^3+z^3=3xyz\)两边关于\(x\)求偏导，得\(3x^2+3z^2\frac{\partialz}{\partialx}=3yz+3xyz_x\)，解得\(\frac{\partialz}{\partialx}=\frac{3xy-x^3}{3xz-y^3}\)。

二、填空题

1.0

解析：对方程\(x^2+y^2+z^2=1\)两边关于\(x\)和\(y\)求偏导，得\(2x+2z\frac{\partialz}{\partialx}=0\)和\(2y+2z\frac{\partialz}{\partialy}=0\)，再对\(x\)求偏导，得\(2+2\left(\frac{\partialz}{\partialx}\right)^2+2z\frac{\partial^2z}{\partialx\partialy}=0\)，由对称性，\(\frac{\partial^2z}{\partialx\partialy}=0\)。

2.\(\frac{2ye^z}{e^{2z}-2y^2}\)

解析：对方程\(e^z=x^2+y^2\)两边关于\(y\)求偏导，得\(e^z\frac{\partialz}{\partialy}=2y\)，解得\(\frac{\partialz}{\partialy}=\frac{2y}{e^z}\)，再对\(y\)求偏导，得\(e^z\frac{\partial^2z}{\partialy^2}+\left(\frac{\partialz}{\partialy}\right)^2e^z=2\)，代入\(\frac{\partialz}{\partialy}\)求解。

3.\(\frac{2}{x^2y^2}\)

解析：对方程\(xyz=1\)两边关于\(x\)求偏导，得\(yz+xyz_x=0\)，解得\(\frac{\partialz}{\partialx}=-\frac{yz}{x^2}\)，再对\(y\)求偏导，得\(z+y\frac{\partialz}{\partialy}+x\frac{\partialz}{\partialx}\frac{\partialy}{\partialy}+xyz_y=0\)，代入\(\frac{\partialz}{\partialx}\)求解。

4.\(-\cos(z)\)

解析：对方程\(\sin(z)=x+y\)两边关于\(x\)求偏导，得\(\cos(z)\frac{\partialz}{\partialx}=1\)，解得\(\frac{\partialz}{\partialx}=\cos(z)\)，再对\(x\)求偏导，得\(-\sin(z)\left(\frac{\partialz}{\partialx}\right)^2+\cos(z)\frac{\partial^2z}{\partialx^2}=0\)，代入\(\frac{\partialz}{\partialx}\)求解。

5.\(-\frac{2x}{3z^2}\)

解析：对方程\(x^2+y^2+z^3=3\)两边关于\(x\)求偏导，得\(2x+3z^2\frac{\partialz}{\partialx}=0\)，解得\(\frac{\partialz}{\partialx}=-\frac{2x}{3z^2}\)，再对\(y\)求偏导，得\(2z\frac{\partialz}{\partialx}+3z^2\frac{\partial^2z}{\partialx\partialy}=0\)，由对称性，\(\frac{\partial^2z}{\partialx\partialy}=-\frac{2x}{3z^2}\)。

6.\(\frac{z}{x^2y^2}\)

解析：对方程\(\ln(z)=x+y\)两边关于\(x\)求偏导，得\(\frac{1}{z}\frac{\partialz}{\partialx}=1\)，解得\(\frac{\partialz}{\partialx}=z\)，再对\(y\)求偏导，得\(\frac{1}{z}\frac{\partialz}{\partialy}=1\)，解得\(\frac{\partialz}{\partialy}=z\)，再对\(x\)求偏导，得\(\frac{1}{z}\frac{\partial^2z}{\partialx\partialy}=0\)，由对称性，\(\frac{\partial^2z}{\partialx\partialy}=\frac{z}{x^2y^2}\)。

7.\(-\frac{2x}{z^2}\)

解析：对方程\(x^2+y^2+z^2=2x+2y+2z\)两边关于\(x\)求偏导，得\(2x+2z\frac{\partialz}{\partialx}=2+2\frac{\partialz}{\partialx}\)，解得\(\frac{\partialz}{\partialx}=\frac{2-x}{z-1}\)，再对\(y\)求偏导，得\(2y+2z\frac{\partialz}{\partialy}=2+2\frac{\partialz}{\partialy}\)，解得\(\frac{\partialz}{\partialy}=\frac{2-y}{z-1}\)，再对\(x\)求偏导，得\(2z\frac{\partial^2z}{\partialx\partialy}+2\frac{\partialz}{\partialy}\frac{\partialz}{\partialx}=0\)，代入\(\frac{\partialz}{\partialx}\)和\(\frac{\partialz}{\partialy}\)求解。

8.\(-\frac{z}{e^{x+y}}\)

解析：对方程\(e^{x+y}=z^2\)两边关于\(y\)求偏导，得\(e^{x+y}\frac{\partialz}{\partialy}=2z\frac{\partialz}{\partialy}\)，解得\(\frac{\partialz}{\partialy}=\frac{z}{e^{x+y}}\)，再对\(y\)求偏导，得\(e^{x+y}\frac{\partial^2z}{\partialy^2}+\left(\frac{\partialz}{\partialy}\right)^2e^{x+y}=0\)，代入\(\frac{\partialz}{\partialy}\)求解。

9.\(\frac{3xy-x^3}{(3xz-y^3)^2}\)

解析：对方程\(x^3+y^3+z^3=3xyz\)两边关于\(x\)求偏导，得\(3x^2+3z^2\frac{\partialz}{\partialx}=3yz+3xyz_x\)，解得\(\frac{\partialz}{\partialx}=\frac{3xy-x^3}{3xz-y^3}\)，再对\(y\)求偏导，得\(3y^2+3z^2\frac{\partialz}{\partialy}=3xz+3xyz_y\)，解得\(\frac{\partialz}{\partialy}=\frac{3xy-y^3}{3xz-x^3}\)，再对\(x\)求偏导，得\(6xy+3z^2\frac{\partial^2z}{\partialx\partialy}+3xyz_y+3xyz_x\frac{\partialz}{\partialy}=3z+3xyz_x\frac{\partialz}{\partialy}\)，代入\(\frac{\partialz}{\partialx}\)和\(\frac{\partialz}{\partialy}\)求解。

10.\(-\frac{2x}{3z^2}\)

解析：对方程\(x^2+y^2+z^3=1\)两边关于\(x\)求偏导，得\(2x+3z^2\frac{\partialz}{\partialx}=0\)，解得\(\frac{\partialz}{\partialx}=-\frac{2x}{3z^2}\)，再对\(y\)求偏导，得\(2z\frac{\partialz}{\partialx}+3z^2\frac{\partial^2z}{\partialx\partialy}=0\)，由对称性，\(\frac{\partial^2z}{\partialx\partialy}=-\frac{2x}{3z^2}\)，再对\(x\)求偏导，得\(2\frac{\partialz}{\partialx}+2z\frac{\partial^2z}{\partialx^2}+3z^2\frac{\partial^2z}{\partialx^2}=0\)，代入\(\frac{\partialz}{\partialx}\)求解。

四、判断题

1.正确

解析：由隐函数求导法则，对\(x^2+y^2+z^2=1\)两边关于\(y\)求偏导，得\(2y+2z\frac{\partialz}{\partialy}=0\)，解得\(\frac{\partialz}{\partialy}=-\frac{y}{z}\)，再对\(x\)求偏导，得\(2z\frac{\partial^2z}{\partialx\partialy}-\frac{y^2}{z^2}\frac{\partialz}{\partialx}=0\)，由对称性，\(\frac{\partial^2z}{\partialx\partialy}=0\)。

2.正确

解析：由隐函数求导法则，对\(e^z=x^2+y^2\)两边关于\(x\)求偏导，得\(e^z\frac{\partialz}{\partialx}=2x\)，解得\(\frac{\partialz}{\partialx}=\frac{2x}{e^z}\)。

3.正确

解析：由隐函数求导法则，对\(xyz=1\)两边关于\(y\)求偏导，得\(xz+xyz_y=0\)，解得\(\frac{\partialz}{\partialy}=-\frac{xz}{xy}=-\frac{z}{x}\)，再对\(x\)求偏导，得\(z+y\frac{\partialz}{\partialy}+x\frac{\partialz}{\partialx}\frac{\partialy}{\partialx}+xyz_x=0\)，代入\(\frac{\partialz}{\partialx}\)和\(\frac{\partialz}{\partialy}\)求解。

4.错误

解析：由隐函数求导法则，对\(\sin(z)=x+y\)两边关于\(y\)求偏导，得\(\cos(z)\frac{\partialz}{\partialy}=1\)，解得\(\frac{\partialz}{\partialy}=\frac{1}{\cos(z)}\)。

5.正确

解析：由隐函数求导法则，对\(x^2+y^2+z^3=3\)两边关于\(y\)求偏导，得\(2y+3z^2\frac{\partialz}{\partialy}=0\)，解得\(\frac{\partialz}{\partialy}=-\frac{2y}{3z^2}\)，再对\(x\)求偏导，得\(2z\frac{\partialz}{\partialx}+3z^2\frac{\partial^2z}{\partialx\partialy}=0\)，由对称性，\(\frac{\partial^2z}{\partialx\partialy}=-\frac{\partial^2z}{\partialy\partialx}\)。

6.错误

解析：由隐函数求导法则，对\(\ln(z)=x+y\)两边关于\(y\)求偏导，得\(\frac{1}{z}\frac{\partialz}{\partialy}=1\)，解得\(\frac{\partialz}{\partialy}=z\)，再对\(x\)求偏导，得\(\frac{1}{z}\frac{\partial^2z}{\partialx\partialy}=\frac{\partialz}{\partialx}\)，由对称性，\(\frac{\partial^2z}{\partialx\partialy}\neq0\)。

7.错误

解析：由隐函数求导法则，对\(x^2+y^2+z^2=2x+2y+2z\)两边关于\(x\)求偏导，得\(2x+2z\frac{\partialz}{\partialx}=2+2\frac{\partialz}{\partialx}\)，解得\(\frac{\partialz}{\partialx}=\frac{2-x}{z-1}\)，在点\((1,1,1)\)处，代入得\(\frac{\partialz}{\partialx}=1\)。

8.正确

解析：由隐函数求导法则，对\(e^{x+y}=z^2\)两边关于\(y\)求偏导，得\(e^{x+y}\frac{\partialz}{\partialy}=2z\frac{\partialz}{\partialy}\)，解得\(\frac{\partialz}{\partialy}=\frac{z}{e^{x+y}}\)，再对\(y\)求偏导，得\(e^{x+y}\frac{\partial^2z}{\partialy^2}+\left(\frac{\partialz}{\partialy}\right)^2e^{x+y}=0\)，代入\(\frac{\partialz}{\partialy}\)求解。

9.错误

解析：由隐函数求导法则，对\(x^3+y^3+z^3=3xyz\)两边关于\(x\)求偏导，得\(3x^2+3z^2\frac{\partialz}{\partialx}=3yz+3xyz_x\)，解得\(\frac{\partialz}{\partialx}=\frac{3xy-x^3}{3xz-y^3}\)，对\(y\)求偏导，得\(3y^2+3z^2\frac{\partialz}{\partialy}=3xz+3xyz_y\)，解得\(\frac{\partialz}{\partialy}=\frac{3xy-y^3}{3xz-x^3}\)，两表达式不同。

10.正确

解析：由隐函数求导法则，对\(x^2+y^2+z^3=1\)两边关于\(x\)求偏导，得\(2x+3z^2\frac{\partialz}{\partialx}=0\)，解得\(\frac{\partialz}{\partialx}=-\frac{2x}{3z^2}\)，再对\(y\)求偏导，得\(2z\frac{\partialz}{\partialx}+3z^2\frac{\partial^2z}{\partialx\partialy}=0\)，由对称性，\(\frac{\partial^2z}{\partialx\partialy}=-\frac{2y}{3z^2}\)，再对\(x\)求偏导，得\(2\frac{\partialz}{\partialx}+2z\frac{\partial^2z}{\partialx^2}+3z^2\frac{\partial^2z}{\partialx^2}=0\)，代入\(\frac{\partialz}{\partialx}\)求解。

五、问答题

1.设\(