導數(shù)中的函數(shù)題目及答案

一、單項選擇題(每題2分,共10題)1.函數(shù)\(y=x^2\)的導數(shù)是(）A.\(2x\)B.\(x\)C.\(2\)D.\(x^3\)2.函數(shù)\(y=\sinx\)的導數(shù)是()A.\(\cosx\)B.\(-\cosx\)C.\(\sinx\)D.\(-\sinx\)3.若\(f(x)=e^x\),則\(f^\prime(x)\)等于()A.\(e^x\)B.\(xe^{x-1}\)C.\(e\)D.\(1\)4.函數(shù)\(y=\lnx\)的導數(shù)是()A.\(\frac{1}{x}\)B.\(x\)C.\(-\frac{1}{x}\)D.\(x^2\)5.已知\(y=3x^3\),則\(y^\prime\)在\(x=1\)處的值為()A.\(9\)B.\(3\)C.\(6\)D.\(1\)6.函數(shù)\(y=\cos2x\)的導數(shù)是()A.\(-2\sin2x\)B.\(2\sin2x\)C.\(-\sin2x\)D.\(\sin2x\)7.若\(f(x)=x^4\),則\(f^\prime(2)\)等于()A.\(32\)B.\(16\)C.\(8\)D.\(4\)8.函數(shù)\(y=\frac{1}{x^2}\)的導數(shù)是()A.\(-\frac{2}{x^3}\)B.\(\frac{2}{x^3}\)C.\(-\frac{1}{x^3}\)D.\(\frac{1}{x^3}\)9.已知\(y=\tanx\),其導數(shù)為()A.\(\sec^2x\)B.\(-\sec^2x\)C.\(\csc^2x\)D.\(-\csc^2x\)10.函數(shù)\(y=5\)的導數(shù)是()A.\(5\)B.\(0\)C.\(1\)D.不存在二、多項選擇題(每題2分,共10題)1.下列函數(shù)求導正確的是(）A.\((x^5)^\prime=5x^4\)B.\((\cosx)^\prime=\sinx\)C.\((e^{2x})^\prime=2e^{2x}\)D.\((\ln3x)^\prime=\frac{1}{x}\)2.以下哪些函數(shù)的導數(shù)與\(x\)有關()A.\(y=x^3+2x\)B.\(y=\cosx\)C.\(y=e^x\)D.\(y=\lnx\)3.函數(shù)\(y=f(x)\)在某點可導,則()A.函數(shù)在該點連續(xù)B.函數(shù)在該點的左導數(shù)等于右導數(shù)C.函數(shù)在該點的切線存在D.函數(shù)在該點的導數(shù)一定不為\(0\)4.求導法則中正確的有()A.\((u+v)^\prime=u^\prime+v^\prime\)B.\((uv)^\prime=u^\primev+uv^\prime\)C.\((\frac{u}{v})^\prime=\frac{u^\primev-uv^\prime}{v^2}(v\neq0)\)D.\((u^n)^\prime=nu^{n-1}\)5.下列函數(shù)中,導數(shù)為偶函數(shù)的有()A.\(y=x^2\)B.\(y=\sinx\)C.\(y=\cosx\)D.\(y=e^x\)6.若函數(shù)\(f(x)\)的導數(shù)\(f^\prime(x)=2x\),則\(f(x)\)可能是()A.\(x^2+1\)B.\(x^2-2\)C.\(2x^2\)D.\(x^2+C\)(\(C\)為常數(shù))7.對于函數(shù)\(y=x^3-3x\),其導數(shù)\(y^\prime\)（）A.可以判斷函數(shù)的單調(diào)性B.令\(y^\prime=0\)可求出函數(shù)的極值點C.\(y^\prime=3x^2-3\)D.\(y^\prime\)在\(x=1\)處的值為\(0\)8.下列函數(shù)求導后是一次函數(shù)的有()A.\(y=x^2\)B.\(y=\frac{1}{x}\)C.\(y=x^3\)D.\(y=\lnx\)9.已知函數(shù)\(f(x)\)在\(x=a\)處可導,\(f^\prime(a)=3\),則()A.函數(shù)\(f(x)\)在\(x=a\)處的切線斜率為\(3\)B.函數(shù)\(f(x)\)在\(x=a\)處的切線方程可設為\(y-f(a)=3(x-a)\)C.函數(shù)\(f(x)\)在\(x=a\)附近的變化率為\(3\)D.函數(shù)\(f(x)\)在\(x=a\)處一定有極值10.若函數(shù)\(y=f(x)\)的導數(shù)\(f^\prime(x)\)的圖象是一條開口向上的拋物線,則()A.\(f(x)\)可能先減后增B.\(f(x)\)有極小值C.\(f^\prime(x)\)可能有兩個零點D.\(f(x)\)一定是二次函數(shù)三、判斷題(每題2分,共10題)1.常數(shù)函數(shù)的導數(shù)為\(0\)。()2.函數(shù)\(y=x^n\)(\(n\)為實數(shù))的導數(shù)一定是\(nx^{n-1}\)。()3.若函數(shù)\(y=f(x)\)在某點的導數(shù)不存在,則函數(shù)在該點不連續(xù)。（)4.函數(shù)\(y=\sin^2x\)的導數(shù)是\(2\sinx\)。()5.曲線\(y=f(x)\)在點\((x_0,f(x_0))\)處的切線方程為\(y-f(x_0)=f^\prime(x_0)(x-x_0)\)。()6.函數(shù)\(y=\ln(-x)\)的導數(shù)是\(\frac{1}{x}\)。()7.兩個函數(shù)和的導數(shù)等于這兩個函數(shù)導數(shù)的和。()8.若\(f^\prime(x)\)恒大于\(0\),則\(f(x)\)是單調(diào)遞增函數(shù)。（)9.函數(shù)\(y=\cos(-x)\)的導數(shù)與\(y=\cosx\)的導數(shù)相同。()10.函數(shù)\(y=x+\frac{1}{x}\)的導數(shù)是\(1-\frac{1}{x^2}\)。(）四、簡答題(每題5分,共4題)1.求函數(shù)\(y=x^3-2x^2+3x-1\)的導數(shù)。答案:根據(jù)求導公式\((x^n)^\prime=nx^{n-1}\),\(y^\prime=3x^2-4x+3\)。2.已知函數(shù)\(y=\sinx\cosx\),求其導數(shù)。答案：利用乘積求導法則\((uv)^\prime=u^\primev+uv^\prime\),\(u=\sinx\),\(u^\prime=\cosx\),\(v=\cosx\),\(v^\prime=-\sinx\),則\(y^\prime=\cos^2x-\sin^2x\)。3.函數(shù)\(y=e^{3x}\)的導數(shù)怎么求?答案：令\(u=3x\),\(y=e^u\),根據(jù)復合函數(shù)求導法則,先對\(e^u\)關于\(u\)求導得\(e^u\),再對\(u\)關于\(x\)求導得\(3\),所以\(y^\prime=3e^{3x}\)。4.求函數(shù)\(y=\ln(x^2+1)\)的導數(shù)。答案:令\(u=x^2+1\),\(y=\lnu\),先對\(\lnu\)關于\(u\)求導得\(\frac{1}{u}\),再對\(u\)關于\(x\)求導得\(2x\),則\(y^\prime=\frac{2x}{x^2+1}\)。五、討論題(每題5分,共4題)1.討論函數(shù)\(y=x^3\)的單調(diào)性與導數(shù)的關系。答案:\(y^\prime=3x^2\geq0\),當\(x\neq0\)時,\(y^\prime>0\),函數(shù)單調(diào)遞增;\(x=0\)時,\(y^\prime=0\)。說明導數(shù)大于\(0\)的區(qū)間函數(shù)遞增,導數(shù)為\(0\)的點不影響整體遞增趨勢。2.結(jié)合導數(shù),分析函數(shù)\(y=\sinx\)在\([0,2\pi]\)上的極值情況。答案：\(y^\prime=\cosx\),令\(y^\prime=0\),在\([0,2\pi]\)上,\(x=\frac{\pi}{2}\)或\(\frac{3\pi}{2}\)。當\(x=\frac{\pi}{2}\),\(y^\prime\)左正右負,\(y\)取極大值\(1\)；當\(x=\frac{3\pi}{2}\),\(y^\prime\)左負右正,\(y\)取極小值\(-1\)。3.探討導數(shù)在實際問題中求最值的應用思路。答案:先建立實際問題的函數(shù)模型,再對函數(shù)求導。令導數(shù)為\(0\)找出可能的極值點,結(jié)合實際問題的定義域,比較極值點與端點處的函數(shù)值,從而確定最大值或最小值。4.說一說函數(shù)\(y=f(x)\)的導數(shù)\(f^\prime(x)\)的正負與函數(shù)圖象的關系。答案:\(f^\prime(x)>0\)時,函數(shù)\(y=f(x)\)圖象上升;\(f^\prime(x)<0\)時,函數(shù)圖象下降;\(f^\prime(x)=0\)處可能是函數(shù)圖象的轉(zhuǎn)折點,即極值點所在位置。答案