導(dǎo)數(shù)題目及答案基礎(chǔ)

一、單項選擇題（每題2分，共10題）1.函數(shù)\(y=x^2\)的導(dǎo)數(shù)是（）A.\(2x\)B.\(x^2\)C.\(2\)D.\(0\)2.函數(shù)\(y=3x\)的導(dǎo)數(shù)為（）A.\(3\)B.\(x\)C.\(3x\)D.\(0\)3.函數(shù)\(y=\sinx\)的導(dǎo)數(shù)是（）A.\(\cosx\)B.\(-\cosx\)C.\(\sinx\)D.\(-\sinx\)4.函數(shù)\(y=e^x\)的導(dǎo)數(shù)是（）A.\(e^x\)B.\(xe^x\)C.\(0\)D.\(1\)5.函數(shù)\(y=\lnx\)的導(dǎo)數(shù)是（）A.\(\frac{1}{x}\)B.\(x\)C.\(\lnx\)D.\(x\lnx\)6.若\(y=5\)，則\(y^\prime\)等于（）A.\(5\)B.\(0\)C.\(1\)D.不存在7.函數(shù)\(y=x^3\)在\(x=1\)處的導(dǎo)數(shù)是（）A.\(1\)B.\(3\)C.\(3x^2\)D.\(0\)8.函數(shù)\(y=\cosx\)在\(x=0\)處的導(dǎo)數(shù)為（）A.\(0\)B.\(1\)C.\(-1\)D.\(\cos0\)9.函數(shù)\(y=2x^2+3x\)的導(dǎo)數(shù)是（）A.\(4x+3\)B.\(2x+3\)C.\(4x\)D.\(3\)10.函數(shù)\(y=\frac{1}{x^2}\)的導(dǎo)數(shù)是（）A.\(-\frac{2}{x^3}\)B.\(\frac{2}{x^3}\)C.\(-\frac{1}{x^3}\)D.\(\frac{1}{x^3}\)二、多項選擇題（每題2分，共10題）1.以下函數(shù)導(dǎo)數(shù)為\(2x\)的有（）A.\(y=x^2+1\)B.\(y=x^2-2\)C.\(y=2x^2\)D.\(y=x^2+c\)（\(c\)為常數(shù)）2.下列函數(shù)導(dǎo)數(shù)正確的是（）A.\(y=\tanx\)的導(dǎo)數(shù)是\(\sec^2x\)B.\(y=\cotx\)的導(dǎo)數(shù)是\(-\csc^2x\)C.\(y=\secx\)的導(dǎo)數(shù)是\(\secx\tanx\)D.\(y=\cscx\)的導(dǎo)數(shù)是\(-\cscx\cotx\)3.函數(shù)\(y=3x^3-2x+1\)的導(dǎo)數(shù)包含以下哪些項（）A.\(9x^2\)B.\(-2\)C.\(0\)D.\(3x^2\)4.若函數(shù)\(y=f(x)\)可導(dǎo)，以下說法正確的是（）A.函數(shù)在某點導(dǎo)數(shù)存在，則在該點連續(xù)B.函數(shù)連續(xù)，則一定可導(dǎo)C.函數(shù)不連續(xù)，則一定不可導(dǎo)D.函數(shù)可導(dǎo)，則一定有極限5.下列函數(shù)中，導(dǎo)數(shù)為偶函數(shù)的有（）A.\(y=x^2\)B.\(y=\cosx\)C.\(y=\sinx\)D.\(y=e^x\)6.函數(shù)\(y=\ln(ax)\)（\(a\neq0\)）的導(dǎo)數(shù)可以是（）A.\(\frac{1}{x}\)B.\(\frac{a}{ax}\)C.\(\frac{1}{ax}\)D.\(\frac{a}{x}\)7.求導(dǎo)公式\((uv)^\prime=u^\primev+uv^\prime\)中，\(u\)、\(v\)可以是（）A.多項式函數(shù)B.三角函數(shù)C.指數(shù)函數(shù)D.對數(shù)函數(shù)8.函數(shù)\(y=x^n\)（\(n\)為實數(shù)）的導(dǎo)數(shù)為\(nx^{n-1}\)，\(n\)可以取（）A.\(1\)B.\(2\)C.\(-1\)D.\(\frac{1}{2}\)9.下列函數(shù)求導(dǎo)運算正確的是（）A.\((x^4)^\prime=4x^3\)B.\((\frac{1}{x})^\prime=-\frac{1}{x^2}\)C.\((e^{-x})^\prime=-e^{-x}\)D.\((\ln2x)^\prime=\frac{1}{x}\)10.導(dǎo)數(shù)為\(0\)的函數(shù)可能是（）A.常數(shù)函數(shù)B.\(y=0\)C.\(y=5\)D.\(y=x\)三、判斷題（每題2分，共10題）1.函數(shù)\(y=x\)的導(dǎo)數(shù)是\(1\)。（）2.函數(shù)\(y=\cosx\)的導(dǎo)數(shù)是\(\sinx\)。（）3.若\(y=f(x)\)在\(x=a\)處不可導(dǎo)，則\(f(x)\)在\(x=a\)處一定不連續(xù)。（）4.常數(shù)的導(dǎo)數(shù)都是\(0\)。（）5.函數(shù)\(y=x^4\)的導(dǎo)數(shù)是\(4x^3\)。（）6.函數(shù)\(y=\lnx\)在\(x=0\)處導(dǎo)數(shù)不存在。（）7.函數(shù)\(y=e^{2x}\)的導(dǎo)數(shù)是\(e^{2x}\)。（）8.兩個函數(shù)和的導(dǎo)數(shù)等于這兩個函數(shù)導(dǎo)數(shù)的和。（）9.函數(shù)\(y=\sin2x\)的導(dǎo)數(shù)是\(2\cos2x\)。（）10.函數(shù)\(y=\frac{1}{x^3}\)的導(dǎo)數(shù)是\(\frac{3}{x^4}\)。（）四、簡答題（每題5分，共4題）1.求函數(shù)\(y=4x^3-2x+1\)的導(dǎo)數(shù)。答案：根據(jù)求導(dǎo)公式\((x^n)^\prime=nx^{n-1}\)，常數(shù)導(dǎo)數(shù)為\(0\)，可得\(y^\prime=4\times3x^2-2=12x^2-2\)。2.求函數(shù)\(y=\sin(3x)\)的導(dǎo)數(shù)。答案：令\(u=3x\)，則\(y=\sinu\)。先對\(\sinu\)關(guān)于\(u\)求導(dǎo)得\(\cosu\)，再對\(u=3x\)關(guān)于\(x\)求導(dǎo)得\(3\)。根據(jù)復(fù)合函數(shù)求導(dǎo)法則，\(y^\prime=3\cos(3x)\)。3.簡述函數(shù)可導(dǎo)與連續(xù)的關(guān)系。答案：函數(shù)可導(dǎo)一定連續(xù)，但連續(xù)不一定可導(dǎo)。即若函數(shù)在某點可導(dǎo)，則在該點一定連續(xù)；而函數(shù)在某點連續(xù)，在該點不一定可導(dǎo)，比如\(y=|x|\)在\(x=0\)處連續(xù)但不可導(dǎo)。4.求函數(shù)\(y=x^2e^x\)的導(dǎo)數(shù)。答案：根據(jù)乘積求導(dǎo)公式\((uv)^\prime=u^\primev+uv^\prime\)，\(u=x^2\)，\(u^\prime=2x\)；\(v=e^x\)，\(v^\prime=e^x\)，所以\(y^\prime=2xe^x+x^2e^x=e^x(x^2+2x)\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=\frac{1}{x}\)在定義域內(nèi)的導(dǎo)數(shù)性質(zhì)。答案：\(y=\frac{1}{x}=x^{-1}\)，導(dǎo)數(shù)\(y^\prime=-x^{-2}=-\frac{1}{x^2}\)。在定義域\((-\infty,0)\cup(0,+\infty)\)內(nèi)，導(dǎo)數(shù)恒小于\(0\)，說明函數(shù)在定義域各區(qū)間單調(diào)遞減，且導(dǎo)數(shù)在\(x=0\)處無定義。2.探討如何根據(jù)導(dǎo)數(shù)判斷函數(shù)的單調(diào)性。答案：若函數(shù)\(y=f(x)\)在某區(qū)間內(nèi)導(dǎo)數(shù)\(f^\prime(x)>0\)，則函數(shù)在該區(qū)間單調(diào)遞增；若\(f^\prime(x)<0\)，則函數(shù)在該區(qū)間單調(diào)遞減；若\(f^\prime(x)=0\)，則該點可能是極值點，函數(shù)單調(diào)性可能改變。3.討論復(fù)合函數(shù)求導(dǎo)的要點。答案：要點是要分清復(fù)合結(jié)構(gòu)，設(shè)出中間變量。先對最外層函數(shù)關(guān)于中間變量求導(dǎo)，再乘以中間變量關(guān)于自