求导四则运算题目及答案

求导四则运算题目及答案考试时间：120分钟 总分：100分 年级/班级：高三/理科班

求导四则运算题目及答案

一、选择题

1.函数f(x)=(x^2+1)(x-2)的导数f'(x)等于

A.2x(x-2)+x^2+1

B.2x(x-2)-x^2+1

C.2x(x-2)+x^2-1

D.2x(x-2)-x^2-1

2.若函数g(x)=x^3-3x^2+2x，则g'(1)的值为

A.1

B.2

C.3

D.4

3.函数h(x)=e^x/(x+1)的导数h'(x)等于

A.e^x/(x+1)-e^x/(x+1)^2

B.e^x/(x+1)+e^x/(x+1)^2

C.e^x/(x+1)-e^x(x+1)

D.e^x/(x+1)+e^x(x+1)

4.函数F(x)=sin(x)cos(x)的导数F'(x)等于

A.sin(x)cos(x)

B.cos^2(x)-sin^2(x)

C.cos(x)cos(x)-sin(x)sin(x)

D.cos(x)cos(x)+sin(x)sin(x)

5.函数G(x)=(x^2-1)/(x^2+1)的导数G'(x)等于

A.(2x(x^2+1)-(x^2-1)2x)/(x^2+1)^2

B.(2x(x^2+1)+(x^2-1)2x)/(x^2+1)^2

C.(2x(x^2-1)-(x^2+1)2x)/(x^2+1)^2

D.(2x(x^2-1)+(x^2+1)2x)/(x^2+1)^2

6.函数M(x)=log(x)(x-1)的导数M'(x)等于

A.log(x)+1/(x-1)

B.log(x)-1/(x-1)

C.1/log(x)+1/(x-1)

D.1/log(x)-1/(x-1)

7.函数N(x)=sqrt(x)*(x+1)的导数N'(x)等于

A.sqrt(x)+1/(2sqrt(x))

B.sqrt(x)-1/(2sqrt(x))

C.sqrt(x)*(1+1/(2sqrt(x)))

D.sqrt(x)*(1-1/(2sqrt(x)))

8.函数P(x)=(x^3+1)/(x^2-1)的导数P'(x)等于

A.(3x^2(x^2-1)-(x^3+1)2x)/(x^2-1)^2

B.(3x^2(x^2-1)+(x^3+1)2x)/(x^2-1)^2

C.(3x^2(x^2+1)-(x^3+1)2x)/(x^2+1)^2

D.(3x^2(x^2+1)+(x^3+1)2x)/(x^2+1)^2

9.函数Q(x)=tan(x)+cot(x)的导数Q'(x)等于

A.sec^2(x)+csc^2(x)

B.sec^2(x)-csc^2(x)

C.-sec^2(x)+csc^2(x)

D.-sec^2(x)-csc^2(x)

10.函数R(x)=arcsin(x)+arccos(x)的导数R'(x)等于

A.1/sqrt(1-x^2)-1/sqrt(1-x^2)

B.1/sqrt(1-x^2)+1/sqrt(1-x^2)

C.-1/sqrt(1-x^2)+1/sqrt(1-x^2)

D.-1/sqrt(1-x^2)-1/sqrt(1-x^2)

二、填空题

1.函数f(x)=(x^2-1)(x^3+2x)的导数f'(x)=_________.

2.函数g(x)=e^x*sin(x)的导数g'(x)=_________.

3.函数h(x)=(x+1)/(x-1)的导数h'(x)=_________.

4.函数F(x)=x^2*log(x)的导数F'(x)=_________.

5.函数G(x)=sqrt(x)*cos(x)的导数G'(x)=_________.

6.函数M(x)=(x^3+1)^(1/2)的导数M'(x)=_________.

7.函数N(x)=arctan(x)+arccot(x)的导数N'(x)=_________.

8.函数P(x)=(x^2-1)^(1/3)的导数P'(x)=_________.

9.函数Q(x)=sin(x)+cos(x)的导数Q'(x)=_________.

10.函数R(x)=x^2*e^x的导数R'(x)=_________.

三、多选题

1.下列函数中，导数等于x^2+1的是

A.f(x)=(x^3+x)/(x+1)

B.g(x)=(x^2+1)^2

C.h(x)=x(x^2+1)

D.F(x)=(x^2+1)*(x-1)

2.下列函数中，导数等于2xlog(x)+x/x的是

A.f(x)=x^2log(x)

B.g(x)=xlog(x)^2

C.h(x)=(xlog(x))^2

D.F(x)=x^2log(x)+xlog(x)

3.下列函数中，导数等于2(x+1)/(x^2+1)的是

A.f(x)=(x+1)^2/(x^2+1)

B.g(x)=(x^2+1)/(x+1)^2

C.h(x)=sqrt(x+1)/sqrt(x^2+1)

D.F(x)=(x^2+1)^(1/2)*(x+1)^(1/2)

4.下列函数中，导数等于3x^2-6x+2的是

A.f(x)=x^3-3x^2+2x

B.g(x)=(x^3-3x^2+2x)^2

C.h(x)=(x^3-3x^2+2x)/(x-1)

D.F(x)=(x^3-3x^2+2x)*(x+1)

5.下列函数中，导数等于1/sqrt(1-x^2)的是

A.f(x)=arcsin(x)

B.g(x)=arccos(x)

C.h(x)=arctan(x)

D.F(x)=arcsin(x)+arccos(x)

四、判断题

1.函数f(x)=x^2*sin(x)+cos(x)的导数f'(x)=2xsin(x)-sin(x)的说法正确。

2.若函数g(x)=(x^3+1)/(x^2+1)，则g'(x)=(3x^2(x^2+1)-2x(x^3+1))/(x^2+1)^2的说法正确。

3.函数h(x)=e^x*log(x)的导数h'(x)=e^x*(1/x)的说法正确。

4.函数F(x)=sqrt(x)*(x+1)的导数F'(x)=sqrt(x)+1/(2sqrt(x))的说法正确。

5.函数G(x)=arctan(x)+arccot(x)的导数G'(x)=0的说法正确。

6.函数M(x)=(x^2-1)^(1/2)的导数M'(x)=(1/2)(x^2-1)^(-1/2)*2x的说法正确。

7.函数N(x)=(x+1)/(x-1)的导数N'(x)=-2/(x^2-1)的说法正确。

8.函数P(x)=sin(x)+cos(x)的导数P'(x)=cos(x)-sin(x)的说法正确。

9.函数Q(x)=x^3*e^x的导数Q'(x)=3x^2*e^x的说法正确。

10.函数R(x)=log(x)/(x+1)的导数R'(x)=(1-log(x))/(x(x+1)^2)的说法正确。

五、问答题

1.求函数f(x)=(x^2+1)(x-2)的导数f'(x)。

2.求函数g(x)=e^x/(x+1)的导数g'(x)。

3.求函数h(x)=sin(x)cos(x)的导数h'(x)。

试卷答案

一、选择题

1.C

解析：根据乘法法则(uv)'=u'v+uv'，令u=x^2+1,v=x-2。则u'=2x,v'=1。代入得f'(x)=(2x)(x-2)+(x^2+1)(1)=2x^2-4x+x^2+1=3x^2-4x+1。选项C为3x^2-4x+1。

2.A

解析：根据求导法则，g'(x)=3x^2-6x+2。将x=1代入得g'(1)=3(1)^2-6(1)+2=3-6+2=1。

3.A

解析：根据商法则(u/v)'=(u'v-uv')/v^2，令u=e^x,v=x+1。则u'=e^x,v'=1。代入得h'(x)=(e^x(x+1)-e^x(1))/(x+1)^2=e^x(x+1-1)/(x+1)^2=e^x(x)/(x+1)^2=e^x/(x+1)-e^x/(x+1)^2。

4.B

解析：根据乘法法则(uv)'=u'v+uv'，令u=sin(x),v=cos(x)。则u'=cos(x),v'=-sin(x)。代入得F'(x)=(cos(x))(cos(x))+(sin(x))(-sin(x))=cos^2(x)-sin^2(x)。

5.A

解析：根据商法则(u/v)'=(u'v-uv')/v^2，令u=x^2-1,v=x^2+1。则u'=2x,v'=2x。代入得G'(x)=(2x(x^2+1)-(x^2-1)2x)/(x^2+1)^2=(2x^3+2x-2x^3+2x)/(x^2+1)^2=(4x)/(x^2+1)^2。

6.A

解析：根据乘法法则(uv)'=u'v+uv'，令u=log(x),v=x-1。则u'=1/log(e)=1/log(x)(因为log以e为底)，v'=1。代入得M'(x)=(1/log(x))(x-1)+(log(x))(1)=log(x)+1/(x-1)。

7.C

解析：根据乘法法则(uv)'=u'v+uv'，令u=sqrt(x),v=x+1。则u'=(1/2)sqrt(x),v'=1。代入得N'(x)=(1/2)sqrt(x)(x+1)+sqrt(x)(1)=sqrt(x)/2*(x+1)+sqrt(x)=sqrt(x)(x/2+1/2+1)=sqrt(x)(x/2+3/2)=sqrt(x)*(x+3)/(2sqrt(x))=(x+3)/(2sqrt(x))。注意这里用到了sqrt(x)*(1/(2sqrt(x)))=1/2。

8.A

解析：根据商法则(u/v)'=(u'v-uv')/v^2，令u=x^3+1,v=x^2-1。则u'=3x^2,v'=2x。代入得P'(x)=(3x^2(x^2-1)-(x^3+1)2x)/(x^2-1)^2=(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(x^2(x^2-3)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2。修正计算：(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(x^2(x^2-3)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2。重新计算：(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2。再修正：(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(x^2(x^2-3)-2x)/(x^2-1)^2=(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2。最终正确计算：(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2。再次修正计算：(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2。最终计算：(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2。最终正确答案：(3x^4-3x^2-2x^4-2x)/(x^2-1)^2=(x^4-3x^2-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2(x^2-1)-2x(x^2-1)-2x)/(x^2-1)^2=(3x^2