復雜函數求導題庫及答案

一、單項選擇題（總共10題，每題2分）1.函數f(x)=x^3-3x^2+2x+1在x=1處的導數是？A.0B.1C.2D.3答案：A2.函數f(x)=e^xsin(x)的導數f'(x)是？A.e^xcos(x)B.e^xsin(x)C.e^x(cos(x)+sin(x))D.e^x(cos(x)-sin(x))答案：C3.函數f(x)=ln(x)/x的導數f'(x)是？A.1-ln(x)B.(1-ln(x))/x^2C.ln(x)-1D.(ln(x)-1)/x^2答案：B4.函數f(x)=sqrt(x)+1/x的導數f'(x)是？A.1/(2sqrt(x))-1/x^2B.sqrt(x)-1/x^2C.1/(2sqrt(x))+1/x^2D.1/(2sqrt(x))+1/x答案：A5.函數f(x)=tan(x)的導數f'(x)是？A.sec^2(x)B.-sec^2(x)C.cot(x)D.-cot(x)答案：A6.函數f(x)=arctan(x)的導數f'(x)是？A.1/(1+x^2)B.1/(1-x^2)C.-1/(1+x^2)D.-1/(1-x^2)答案：A7.函數f(x)=sin^2(x)的導數f'(x)是？A.2sin(x)cos(x)B.sin(2x)C.2sin(x)D.2cos(x)答案：B8.函數f(x)=cos^3(x)的導數f'(x)是？A.-3cos^2(x)sin(x)B.3cos^2(x)sin(x)C.-2cos^2(x)sin(x)D.2cos^2(x)sin(x)答案：A9.函數f(x)=x^2e^x的導數f'(x)是？A.2xe^xB.x^2e^xC.(2x+x^2)e^xD.(2x+1)e^x答案：D10.函數f(x)=(x^2+1)/(x-1)的導數f'(x)是？A.(2x(x-1)-(x^2+1))/(x-1)^2B.(2x(x-1)+(x^2+1))/(x-1)^2C.(2x(x+1)-(x^2+1))/(x-1)^2D.(2x(x+1)+(x^2+1))/(x-1)^2答案：A二、多項選擇題（總共10題，每題2分）1.下列函數中，哪些在x=0處可導？A.f(x)=x^2B.f(x)=|x|C.f(x)=x^3D.f(x)=1/x答案：AC2.下列函數中，哪些的導數在x=0處存在？A.f(x)=sin(x)B.f(x)=cos(x)C.f(x)=tan(x)D.f(x)=sec(x)答案：AB3.下列函數中，哪些是可導函數？A.f(x)=e^xB.f(x)=ln(x)C.f(x)=sqrt(x)D.f(x)=1/x答案：ABCD4.下列函數中，哪些的導數是連續(xù)函數？A.f(x)=x^2B.f(x)=sin(x)C.f(x)=cos(x)D.f(x)=tan(x)答案：ABC5.下列函數中，哪些是奇函數？A.f(x)=x^3B.f(x)=sin(x)C.f(x)=cos(x)D.f(x)=tan(x)答案：ABD6.下列函數中，哪些是偶函數？A.f(x)=x^2B.f(x)=cos(x)C.f(x)=sin(x)D.f(x)=tan(x)答案：AB7.下列函數中，哪些的導數是偶函數？A.f(x)=x^2B.f(x)=sin(x)C.f(x)=cos(x)D.f(x)=tan(x)答案：AC8.下列函數中，哪些的導數是奇函數？A.f(x)=x^3B.f(x)=sin(x)C.f(x)=cos(x)D.f(x)=tan(x)答案：ABD9.下列函數中，哪些是周期函數？A.f(x)=sin(x)B.f(x)=cos(x)C.f(x)=tan(x)D.f(x)=e^x答案：ABC10.下列函數中，哪些的導數是周期函數？A.f(x)=sin(x)B.f(x)=cos(x)C.f(x)=tan(x)D.f(x)=e^x答案：ABC三、判斷題（總共10題，每題2分）1.函數f(x)=x^2在x=0處的導數是0。答案：正確2.函數f(x)=e^x在x=0處的導數是1。答案：正確3.函數f(x)=sin(x)在x=π處的導數是0。答案：正確4.函數f(x)=cos(x)在x=π/2處的導數是0。答案：正確5.函數f(x)=ln(x)在x=1處的導數是1。答案：正確6.函數f(x)=sqrt(x)在x=4處的導數是1/4。答案：錯誤7.函數f(x)=tan(x)在x=π/4處的導數是1。答案：正確8.函數f(x)=arctan(x)在x=0處的導數是1。答案：錯誤9.函數f(x)=sin^2(x)在x=π/2處的導數是1。答案：錯誤10.函數f(x)=cos^3(x)在x=π/2處的導數是0。答案：正確四、簡答題（總共4題，每題5分）1.函數f(x)=x^3-3x^2+2x+1的導數f'(x)是什么？答案：f'(x)=3x^2-6x+22.函數f(x)=e^xsin(x)的二階導數f''(x)是什么？答案：f''(x)=e^x(sin(x)+cos(x))3.函數f(x)=ln(x)/x的導數f'(x)是什么？答案：f'(x)=(1-ln(x))/x^24.函數f(x)=sqrt(x)+1/x的導數f'(x)是什么？答案：f'(x)=1/(2sqrt(x))-1/x^2五、討論題（總共4題，每題5分）1.討論函數f(x)=x^3-3x^2+2x+1的單調性和極值點。答案：f'(x)=3x^2-6x+2，令f'(x)=0，解得x=1±sqrt(1/3)。在x=1-sqrt(1/3)和x=1+sqrt(1/3)處取得極值，分別為極大值和極小值。2.討論函數f(x)=e^xsin(x)的周期性和導數的性質。答案：f(x)=e^xsin(x)是周期函數，周期為2π。導數f'(x)=e^x(sin(x)+cos(x))也是周期函數，周期為2π。3.討論函數f(x)=ln(x)/x的極值點和單調性。答案：f'(x)=(1-ln(x))/x^2，令f'(x)=0，解得x=