中學(xué)導(dǎo)數(shù)考試題庫及答案

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.\(y'=x\)B.\(y'=2x\)C.\(y'=2x^{2}\)D.\(y'=\frac{1}{2}x\)答案：B2.函數(shù)\(y=\sinx\)的導(dǎo)數(shù)是（）A.\(y'=\cosx\)B.\(y'=-\cosx\)C.\(y'=\sinx\)D.\(y'=-\sinx\)答案：A3.函數(shù)\(y=e^{x}\)的導(dǎo)數(shù)是（）A.\(y'=\frac{1}{e^{x}}\)B.\(y'=e^{x}\)C.\(y'=-e^{x}\)D.\(y'=\lnx\)答案：B4.函數(shù)\(y=\lnx\)的導(dǎo)數(shù)是（）A.\(y'=\frac{1}{x}\)B.\(y'=-\frac{1}{x}\)C.\(y'=\frac{1}{\lnx}\)D.\(y'=x\)答案：A5.函數(shù)\(y=3x^{3}\)的導(dǎo)數(shù)是（）A.\(y'=9x^{2}\)B.\(y'=6x^{2}\)C.\(y'=3x^{2}\)D.\(y'=12x^{2}\)答案：A6.若\(y=f(x)\)在\(x=a\)處可導(dǎo)，則\(\lim\limits_{\Deltax\rightarrow0}\frac{f(a+\Deltax)-f(a)}{\Deltax}\)等于（）A.\(f'(a)\)B.\(f(a)\)C.\(f'(a+\Deltax)\)D.\(f(a+\Deltax)\)答案：A7.函數(shù)\(y=(x+1)^{2}\)的導(dǎo)數(shù)\(y'\)在\(x=1\)處的值為（）A.4B.3C.2D.1答案：A8.函數(shù)\(y=\cos2x\)的導(dǎo)數(shù)是（）A.\(y'=-2\sin2x\)B.\(y'=2\sin2x\)C.\(y'=-2\cos2x\)D.\(y'=2\cos2x\)答案：A9.設(shè)\(y=f(x)\)，如果\(f'(x_{0})=0\)，則\(x_{0}\)是函數(shù)\(y=f(x)\)的（）A.極大值點B.極小值點C.駐點D.最大值點答案：C10.函數(shù)\(y=x^{3}-3x^{2}+1\)的單調(diào)遞增區(qū)間是（）A.\((-\infty,0)\cup(2,+\infty)\)B.\((0,2)\)C.\((-\infty,1)\cup(3,+\infty)\)D.\((1,3)\)答案：A二、多項選擇題（每題2分，共10題）1.下列函數(shù)中，導(dǎo)數(shù)為\(2x\)的函數(shù)是（）A.\(y=x^{2}\)B.\(y=2x^{2}-x\)C.\(y=x^{2}+1\)D.\(y=\frac{1}{2}x^{2}+x\)答案：AC2.函數(shù)\(y=\sin2x\)的導(dǎo)數(shù)與下列哪些函數(shù)的導(dǎo)數(shù)相同（）A.\(y=2\cos2x\)B.\(y=-2\cos2x\)C.\(y=\cos2x\)D.\(y=-\cos2x\)答案：BD3.對于函數(shù)\(y=e^{2x}\)，下列說法正確的是（）A.其導(dǎo)數(shù)為\(y'=2e^{2x}\)B.其導(dǎo)數(shù)為\(y'=e^{2x}\)C.在\(x=0\)處的導(dǎo)數(shù)為2D.在\(x=0\)處的導(dǎo)數(shù)為1答案：AC4.若函數(shù)\(y=f(x)\)在區(qū)間\((a,b)\)內(nèi)可導(dǎo)，則（）A.若\(f'(x)>0\)，則\(y=f(x)\)在\((a,b)\)內(nèi)單調(diào)遞增B.若\(f'(x)<0\)，則\(y=f(x)\)在\((a,b)\)內(nèi)單調(diào)遞減C.若\(f'(x)=0\)，則\(y=f(x)\)在\((a,b)\)內(nèi)為常數(shù)函數(shù)D.若\(f'(x)\)在\((a,b)\)內(nèi)恒為正，則\(y=f(x)\)在\((a,b)\)內(nèi)單調(diào)遞增答案：ABD5.下列函數(shù)在\(x=0\)處導(dǎo)數(shù)存在的有（）A.\(y=\vertx\vert\)B.\(y=x^{2}\)C.\(y=\sinx\)D.\(y=\frac{1}{x}\)答案：BC6.設(shè)\(y=f(x)\)，\(y=g(x)\)都可導(dǎo)，則\((f(x)+g(x))'\)等于（）A.\(f'(x)+g'(x)\)B.\(f(x)+g'(x)\)C.\(f'(x)+g(x)\)D.\(f(x)+g(x)\)答案：A7.設(shè)\(y=f(x)\)可導(dǎo)，則\((cf(x))'\)（\(c\)為常數(shù)）等于（）A.\(cf'(x)\)B.\(c+f'(x)\)C.\(f(x)+c\)D.\(c\)答案：A8.函數(shù)\(y=\frac{1}{x^{2}}\)的導(dǎo)數(shù)是（）A.\(y'=-\frac{2}{x^{3}}\)B.\(y'=\frac{2}{x^{3}}\)C.\(y'=-2x^{-3}\)D.\(y'=2x^{-3}\)答案：AC9.函數(shù)\(y=\sqrt{x}\)的導(dǎo)數(shù)是（）A.\(y'=\frac{1}{2\sqrt{x}}\)B.\(y'=\frac{1}{\sqrt{x}}\)C.\(y'=2\sqrt{x}\)D.\(y'=\frac{\sqrt{x}}{2}\)答案：A10.設(shè)\(y=f(u)\)，\(u=g(x)\)都可導(dǎo)，則\(y=f(g(x))\)的導(dǎo)數(shù)\(y'\)等于（）A.\(f'(u)g'(x)\)B.\(f(g(x))g'(x)\)C.\(f'(g(x))g'(x)\)D.\(f(g(x))+g'(x)\)答案：C三、判斷題（每題2分，共10題）1.函數(shù)\(y=x^{3}\)的導(dǎo)數(shù)是\(y'=3x^{2}\)。（）答案：正確2.函數(shù)\(y=\ln(x+1)\)的導(dǎo)數(shù)是\(y'=\frac{1}{x+1}\)。（）答案：正確3.函數(shù)\(y=\cosx\)在\(x=0\)處的導(dǎo)數(shù)為0。（）答案：錯誤4.若\(y=f(x)\)在\(x=a\)處不可導(dǎo)，則\(y=f(x)\)在\(x=a\)處不連續(xù)。（）答案：錯誤5.函數(shù)\(y=e^{x}\)的導(dǎo)數(shù)恒為\(e^{x}\)，所以\(y=e^{x}\)沒有極值點。（）答案：正確6.函數(shù)\(y=\frac{1}{x}\)在\((0,+\infty)\)上單調(diào)遞減。（）答案：正確7.函數(shù)\(y=x^{2}\sinx\)的導(dǎo)數(shù)為\(y'=2x\sinx+x^{2}\cosx\)。（）答案：正確8.若\(f'(x_{0})=0\)，則\(x_{0}\)一定是函數(shù)\(y=f(x)\)的極值點。（）答案：錯誤9.函數(shù)\(y=\sqrt{x^{2}}\)與\(y=\vertx\vert\)是同一個函數(shù)，它們的導(dǎo)數(shù)也相同。（）答案：錯誤10.函數(shù)\(y=2x^{3}-3x^{2}+1\)的二階導(dǎo)數(shù)\(y''=12x-6\)。（）答案：正確四、簡答題（每題5分，共4題）1.求函數(shù)\(y=x^{4}-2x^{3}+3x^{2}-4x+5\)的導(dǎo)數(shù)。答案：\(y'=4x^{3}-6x^{2}+6x-4\)。2.求函數(shù)\(y=\sin3x\)在\(x=\frac{\pi}{3}\)處的導(dǎo)數(shù)。答案：\(y'=3\cos3x\)，當\(x=\frac{\pi}{3}\)時，\(y'=-3\)。3.求函數(shù)\(y=\ln(2x+1)\)的導(dǎo)數(shù)。答案：\(y'=\frac{2}{2x+1}\)。4.已知函數(shù)\(y=f(x)\)在\(x=a\)處可導(dǎo)，且\(f(a)=3\)，\(f'(a)=2\)，求\(\lim\limits_{\Deltax\rightarrow0}\frac{f(a+\Deltax)-3}{\Deltax}\)。答案：\(\lim\limits_{\Deltax\rightarrow0}\frac{f(a+\Deltax)-3}{\Deltax}=\lim\limits_{\Deltax\rightarrow0}\frac{f(a+\Deltax)-f(a)}{\Deltax}=f'(a)=2\)。五、討論題（每題5分，共4題）1.討論函數(shù)\(y=x^{3}-3x\)的單調(diào)性。答案：\(y'=3x^{2}-3=3(x+1)(x-1)\)。當\(x<-1\)或\(x>1\)時，\(y'>0\)，函數(shù)單調(diào)遞增；當\(-1<x<1\)時，\(y'<0\)，函數(shù)單調(diào)遞減。2.求函數(shù)\(y=e^{x}-x-1\)的極值。答案：\(y'=e^{x}-1\)，令\(y'=0\)，得\(x=0\)。當\(x<0\)時，\(y'<0\)；當\(x>0\)時，\(y'>0\)。所以\(x=0\)時函數(shù)有極小值\(y(0)=0\)。3.設(shè)函數(shù)\(y=f(x)\)的導(dǎo)數(shù)\(y'=(x-1)(x-2)\)，討論函數(shù)\(y=f(x)\)