1、2020/9/23,1,第二章 一元函數(shù)微分學,第一節(jié) 導數(shù),2020/9/23,2,一、問題的提出,1、變速直線運動的速度問題,取極限得,瞬時速度,2020/9/23,3,2、切線問題,切線：割線的極限,播放,M,N,T,割線MN繞點M旋轉而趨向極限位置MT,直線MT就稱為曲線C在點M處的切線.,2020/9/23,4,Q,2020/9/23,5,二、導數(shù)的定義,1、定義,2020/9/23,6,導數(shù)定義其它常見形式：,即,2020/9/23,7,右導數(shù)：,單側導數(shù),左導數(shù)：,判斷函數(shù)在某一點可導的充分必要條件：,2020/9/23,8,例1,解,2020/9/23,9,1）,2、導函數(shù),2

2、）,2020/9/23,10,3）,注：,2020/9/23,11,三、導數(shù)的幾何意義,切線方程為,法線方程為,2020/9/23,12,例2,解,根據(jù)導數(shù)的幾何意義，得切線斜率為：,故切線方程為,法線方程為,2020/9/23,13,四、可導與連續(xù)的關系,結論：可導的函數(shù)一定是連續(xù)的,證,2020/9/23,14,例3,解,2020/9/23,15,思考判斷題,2020/9/23,16,一、按定義求導數(shù),例1,解,第二節(jié) 初等函數(shù)的導數(shù),2020/9/23,17,例2,解,2020/9/23,18,例3,解,更一般地,2020/9/23,19,例4,解,2020/9/23,20,例5,解,2

3、020/9/23,21,例6,解,2020/9/23,22,例7,解,2020/9/23,23,例8,解,2020/9/23,24,二、和、差、積、商的求導法則,定理,2020/9/23,25,證(1),則,2020/9/23,26,證(2),則,2020/9/23,27,證(3),則,2020/9/23,28,2020/9/23,29,推論,2020/9/23,30,例1,解,例2,解,2020/9/23,31,解,例3,2020/9/23,32,例4,解,2020/9/23,33,解,例5,2020/9/23,34,例6,解,例7,解,2020/9/23,35,思考題,求曲線 上與 軸平行

4、的切線方程.,2020/9/23,36,思考題解答,令,切點為,所求切線方程為,和,2020/9/23,37,（一）反函數(shù)的導數(shù),定理,即 反函數(shù)的導數(shù)等于原函數(shù)導數(shù)的倒數(shù).,二、反函數(shù)的導數(shù)、復合函數(shù)的求導法則,2020/9/23,38,證,于是有,2020/9/23,39,2020/9/23,40,2020/9/23,41,2020/9/23,42,2020/9/23,43,（二）復合函數(shù)的求導法則（鏈式法則）,定理,把特殊點 換成一般點 ，有,2020/9/23,44,推廣,例5,解,令,則有,2020/9/23,45,例6,解,2020/9/23,46,例7,解,2020/9/23,4

5、7,例8,解,2020/9/23,48,例9,解 這里僅考慮 x 為正數(shù)的情況,2020/9/23,49,例10,解,2020/9/23,50,例11,解,2020/9/23,51,思考題,冪函數(shù)在其定義域內(nèi)（ ）.,2020/9/23,52,（一）隱函數(shù)的導數(shù),三、隱函數(shù)的導數(shù) 對數(shù)求導方法 高階導數(shù),例如：,2020/9/23,53,隱函數(shù)求導法則:,用復合函數(shù)求導法則直接對方程兩邊求導.,問題:隱函數(shù)不易顯化或不能顯化時如何求導?,隱函數(shù)的顯化；,若,若無法寫成上面的形式，則認為不能顯化,2020/9/23,54,例1,解,解得,2020/9/23,55,例2,解,解得,2020/9/23,56,隱函數(shù)求導法可用來建立反函數(shù)的導數(shù),例3,解,2020/9/23,57,例4,解,2020/9/23,58,（二）對數(shù)求導法,觀察函數(shù),方法:,先在方程兩邊取對數(shù), 然后利用隱函數(shù)求導法.,適用范圍:,2020/9/23,59,例1,解,等式兩邊取對數(shù)得,2020/9/23,60,例2,解,等式兩邊取對數(shù)得,2020/9/23,61,問題:變速直線運動的加速度.,定義,（三）高階導數(shù),2020/9/23,62,記作:,三階導數(shù)的導數(shù)稱為四階導數(shù),記作:,二階導數(shù)的導數(shù)稱為三階導數(shù),記作:,2020/9/23,63,例1,解,二階和二階以上的