新高考數(shù)學(xué)一輪復(fù)習(xí)講義 第14講 導(dǎo)數(shù)的概念及其意義、導(dǎo)數(shù)的運(yùn)算（原卷版）

第14講導(dǎo)數(shù)的概念及其意義、導(dǎo)數(shù)的運(yùn)算（精講）題型目錄一覽①導(dǎo)數(shù)的定義②導(dǎo)數(shù)的運(yùn)算③導(dǎo)數(shù)中的切線問題Ⅰ-求在曲線上一點(diǎn)的切線方程④導(dǎo)數(shù)中的切線問題Ⅱ-求過一點(diǎn)的切線方程⑤導(dǎo)數(shù)中的切線問題Ⅲ-求參數(shù)的值(范圍)★【文末附錄-導(dǎo)數(shù)的概念及其意義和導(dǎo)數(shù)的運(yùn)算思維導(dǎo)圖】一、知識點(diǎn)梳理一、知識點(diǎn)梳理一、導(dǎo)數(shù)的概念和幾何性質(zhì)1.概念函數(shù)SKIPIF1<0在SKIPIF1<0處瞬時變化率是SKIPIF1<0，我們稱它為函數(shù)SKIPIF1<0在SKIPIF1<0處的導(dǎo)數(shù),記作SKIPIF1<0或SKIPIF1<0．注：增量SKIPIF1<0可以是正數(shù)，也可以是負(fù)，但是不可以等于0．SKIPIF1<0的意義：SKIPIF1<0與0之間距離要多近有多近，即SKIPIF1<0可以小于給定的任意小的正數(shù)；2.幾何意義函數(shù)SKIPIF1<0在SKIPIF1<0處的導(dǎo)數(shù)SKIPIF1<0的幾何意義即為函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線的斜率．二、導(dǎo)數(shù)的運(yùn)算1.求導(dǎo)的基本公式基本初等函數(shù)導(dǎo)函數(shù)SKIPIF1<0（SKIPIF1<0為常數(shù)）SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<02.導(dǎo)數(shù)的四則運(yùn)算法則（1）函數(shù)和差求導(dǎo)法則：SKIPIF1<0；（2）函數(shù)積的求導(dǎo)法則：SKIPIF1<0；（3）函數(shù)商的求導(dǎo)法則：SKIPIF1<0，則SKIPIF1<0．3.復(fù)合函數(shù)求導(dǎo)數(shù)復(fù)合函數(shù)SKIPIF1<0的導(dǎo)數(shù)和函數(shù)SKIPIF1<0，SKIPIF1<0的導(dǎo)數(shù)間關(guān)系為SKIPIF1<0：【常用結(jié)論】1.在點(diǎn)的切線方程切線方程SKIPIF1<0的計算：函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程為SKIPIF1<0，抓住關(guān)鍵SKIPIF1<0．2.過點(diǎn)的切線方程設(shè)切點(diǎn)為SKIPIF1<0，則斜率SKIPIF1<0，過切點(diǎn)的切線方程為：SKIPIF1<0，又因?yàn)榍芯€方程過點(diǎn)SKIPIF1<0，所以SKIPIF1<0然后解出SKIPIF1<0的值．（SKIPIF1<0有幾個值，就有幾條切線）二、題型分類精講二、題型分類精講題型一導(dǎo)數(shù)的定義策略方法對所給函數(shù)式經(jīng)過添項、拆項等恒等變形與導(dǎo)數(shù)定義結(jié)構(gòu)相同，然后根據(jù)導(dǎo)數(shù)定義直接寫出.【典例1】已知函數(shù)SKIPIF1<0在SKIPIF1<0處的導(dǎo)數(shù)SKIPIF1<0，則SKIPIF1<0（

）．A．SKIPIF1<0 B．1 C．SKIPIF1<0 D．SKIPIF1<0【題型訓(xùn)練】一、單選題1．設(shè)SKIPIF1<0為SKIPIF1<0上的可導(dǎo)函數(shù)，且SKIPIF1<0，則曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線斜率為（

）A．2 B．-1 C．1 D．SKIPIF1<02．已知函數(shù)SKIPIF1<0的導(dǎo)函數(shù)是SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0（）A．SKIPIF1<0 B．1 C．2 D．4二、填空題3．已知函數(shù)SKIPIF1<0，則SKIPIF1<0______．題型二導(dǎo)數(shù)的運(yùn)算策略方法對所給函數(shù)求導(dǎo)，其方法是利用和、差、積、商及復(fù)合函數(shù)求導(dǎo)法則，直接轉(zhuǎn)化為基本函數(shù)求導(dǎo)問題.【典例1】求下列函數(shù)的導(dǎo)數(shù)．(1)SKIPIF1<0；(2)SKIPIF1<0；(3)SKIPIF1<0(4)SKIPIF1<0；(5)SKIPIF1<0（SKIPIF1<0為常數(shù)）；(6)SKIPIF1<0.【題型訓(xùn)練】一、解答題1．下列函數(shù)的導(dǎo)函數(shù)（1）SKIPIF1<0；（2）SKIPIF1<0；（3）SKIPIF1<0；（4）SKIPIF1<0.2．求下列函數(shù)的導(dǎo)數(shù)．(1)SKIPIF1<0；(2)SKIPIF1<0；(3)SKIPIF1<0(4)SKIPIF1<0；3．求下列函數(shù)的導(dǎo)數(shù)：(1)SKIPIF1<0；(2)SKIPIF1<0；(3)SKIPIF1<0；(4)SKIPIF1<0；(5)SKIPIF1<0；(6)SKIPIF1<0．題型三導(dǎo)數(shù)中的切線問題Ⅰ-求在曲線上一點(diǎn)的切線方程策略方法已知切點(diǎn)A(x0，f(x0))求切線方程，可先求該點(diǎn)處的導(dǎo)數(shù)值f′(x0)，再根據(jù)y－f(x0)＝f′(x0)(x－x0)求解．【典例1】設(shè)曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線與直線SKIPIF1<0平行，則實(shí)數(shù)SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【題型訓(xùn)練】一、單選題1．已知函數(shù)SKIPIF1<0，則SKIPIF1<0的圖象在SKIPIF1<0處的切線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．已知函數(shù)SKIPIF1<0，若SKIPIF1<0的圖象在SKIPIF1<0處的切線與坐標(biāo)軸圍成的三角形的面積為1，則SKIPIF1<0（

）A．SKIPIF1<0 B．2 C．±2 D．SKIPIF1<03．已知SKIPIF1<0為實(shí)數(shù)，函數(shù)SKIPIF1<0是偶函數(shù)，則曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、填空題4．已知曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線與曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線互相垂直，則SKIPIF1<0________．5．已知函數(shù)SKIPIF1<0的圖象在SKIPIF1<0處的切線在y軸上的截距為2，則實(shí)數(shù)SKIPIF1<0____________．6．已知函數(shù)SKIPIF1<0，則曲線SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線方程為__________.7．已知函數(shù)SKIPIF1<0，直線SKIPIF1<0，SKIPIF1<0是SKIPIF1<0的兩條切線，SKIPIF1<0，SKIPIF1<0相交于點(diǎn)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0點(diǎn)橫坐標(biāo)的取值范圍是________．三、解答題8．已知函數(shù)SKIPIF1<0，其中SKIPIF1<0.若曲線SKIPIF1<0在SKIPIF1<0處的切線過點(diǎn)SKIPIF1<0，求SKIPIF1<0的值；題型四導(dǎo)數(shù)中的切線問題Ⅱ-求過一點(diǎn)的切線方程策略方法設(shè)切點(diǎn)為SKIPIF1<0，則斜率SKIPIF1<0，過切點(diǎn)的切線方程為：SKIPIF1<0，又因?yàn)榍芯€方程過點(diǎn)SKIPIF1<0，所以SKIPIF1<0然后解出SKIPIF1<0的值【典例1】過原點(diǎn)且與函數(shù)SKIPIF1<0圖像相切的直線方程是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【題型訓(xùn)練】一、單選題1．若直線SKIPIF1<0為曲線SKIPIF1<0的一條切線，則實(shí)數(shù)k的值是（

）A．e B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．過坐標(biāo)原點(diǎn)作曲線SKIPIF1<0的切線，則切線方程為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．若過點(diǎn)SKIPIF1<0可以作曲線SKIPIF1<0的兩條切線，則（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．過坐標(biāo)原點(diǎn)作曲線SKIPIF1<0的切線，則切線有（

）條A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、填空題5．過坐標(biāo)原點(diǎn)作曲線SKIPIF1<0的切線，則切點(diǎn)的橫坐標(biāo)為___________.6．已知直線SKIPIF1<0與曲線SKIPIF1<0相切，則SKIPIF1<0_________.7．過點(diǎn)SKIPIF1<0作曲線SKIPIF1<0的兩條切線，則這兩條切線的斜率之和為______.8．若曲線SKIPIF1<0有兩條過坐標(biāo)原點(diǎn)的切線，則實(shí)a的取值范圍為______.題型五導(dǎo)數(shù)中的切線問題Ⅲ-求參數(shù)的值(范圍)策略方法1．利用導(dǎo)數(shù)的幾何意義求參數(shù)的基本方法利用切點(diǎn)的坐標(biāo)、切線的斜率、切線的方程等得到關(guān)于參數(shù)的方程(組)或者參數(shù)滿足的不等式(組)，進(jìn)而求出參數(shù)的值或取值范圍．2．求解與導(dǎo)數(shù)的幾何意義有關(guān)問題時應(yīng)注意的兩點(diǎn)(1)注意曲線上橫坐標(biāo)的取值范圍．(2)謹(jǐn)記切點(diǎn)既在切線上又在曲線上．【典例1】已知函數(shù)SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線為SKIPIF1<0，則SKIPIF1<0的值為（

）A．1 B．2 C．3 D．4【題型訓(xùn)練】一、單選題1．已知曲線SKIPIF1<0在點(diǎn)P處的切線與直線SKIPIF1<0垂直，則點(diǎn)P的橫坐標(biāo)為（

）A．1 B．SKIPIF1<0 C．2 D．SKIPIF1<02．已知函數(shù)SKIPIF1<0與SKIPIF1<0的圖象在SKIPIF1<0處有相同的切線，則SKIPIF1<0（

）A．0 B．SKIPIF1<0 C．1 D．SKIPIF1<0或13．若點(diǎn)P是函數(shù)SKIPIF1<0任意一點(diǎn)，則點(diǎn)P到直線SKIPIF1<0的最小距離為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<04．動直線SKIPIF1<0分別與直線SKIPIF1<