醫(yī)用高數(shù) 2初等函數(shù)導(dǎo)數(shù)

第二節(jié)、初等函數(shù)的導(dǎo)數(shù)

一、按定義求導(dǎo)數(shù)步驟:例1解6/20/2023例2解6/20/2023例3解更一般地例如,6/20/2023例4解6/20/2023例5解6/20/2023例6解6/20/2023二、四則運(yùn)算的求導(dǎo)法則定理并且可導(dǎo)處也在點(diǎn)分母不為零們的和、差、積、商則它處可導(dǎo)在點(diǎn)如果函數(shù),)(,)(),(xxxvxu6/20/2023推論6/20/2023三、反函數(shù)的導(dǎo)數(shù)即反函數(shù)的導(dǎo)數(shù)等于直接函數(shù)導(dǎo)數(shù)的倒數(shù).定理.)(1)(,)(,0)()(yxfIxfyyIyxxyj￠=￠=1j￠j=且有內(nèi)也可導(dǎo)在對(duì)應(yīng)區(qū)間那末它的反函數(shù)且內(nèi)單調(diào)、可導(dǎo)在某區(qū)間如果函數(shù)6/20/2023例1解同理可得6/20/2023例2解特別地6/20/2023四、復(fù)合函數(shù)的求導(dǎo)法則即因變量對(duì)自變量求導(dǎo),等于因變量對(duì)中間變量求導(dǎo),乘以中間變量對(duì)自變量求導(dǎo).(鏈?zhǔn)椒▌t)定理).()(,)]([,)()(,)(0000000xufdxdyxxfyxuufyxxuxxj￠*￠=j=j==j==且其導(dǎo)數(shù)為可導(dǎo)在點(diǎn)則復(fù)合函數(shù)可導(dǎo)在點(diǎn)而可導(dǎo)在點(diǎn)如果函數(shù)6/20/2023推廣例1解6/20/2023例2解例3解6/20/2023例4解例5解6/20/2023五、隱函數(shù)的導(dǎo)數(shù)定義:隱函數(shù)的顯化問(wèn)題:隱函數(shù)不易顯化或不能顯化如何求導(dǎo)?6/20/2023用復(fù)合函數(shù)求導(dǎo)法則直接對(duì)方程兩邊求導(dǎo).隱函數(shù)求導(dǎo)法則:在求導(dǎo)過(guò)程中注意y是x的函數(shù)，利用復(fù)合函數(shù)的求導(dǎo)法則便可得到所求函數(shù)的導(dǎo)數(shù)（導(dǎo)數(shù)結(jié)果中允許含有y）。注意：6/20/2023例1解解得6/20/2023例2解所求切線方程為顯然通過(guò)原點(diǎn).6/20/2023六、對(duì)數(shù)求導(dǎo)法觀察函數(shù)方法:先在方程兩邊取對(duì)數(shù),然后利用隱函數(shù)的求導(dǎo)方法求出導(dǎo)數(shù).--------對(duì)數(shù)求導(dǎo)法適用范圍:6/20/2023例1解等式兩邊取對(duì)數(shù)得6/20/2023例2解等式兩邊取對(duì)數(shù)得6/20/2023一般地6/20/2023七、初等函數(shù)的導(dǎo)數(shù)注意:分段函數(shù)求導(dǎo)時(shí),分界點(diǎn)導(dǎo)數(shù)用左右導(dǎo)數(shù)求.6/20/20231.常數(shù)和基本初等函數(shù)的導(dǎo)數(shù)公式6/20/20232.函數(shù)的和、差、積、商的求導(dǎo)法則設(shè))(),(xvvxuu==可導(dǎo)，則(3)vuvuuv￠+￠=￠)(,

(4))0()(21￠-￠=￠vvvuvuvu.uccu￠=￠)((是常數(shù))(2)vuvu

￠￠=￠

)(,(1)6/20/20233.復(fù)合函數(shù)的求導(dǎo)法則利用上述公式及法則初等函數(shù)求導(dǎo)問(wèn)題可完全解決.注意:初等函數(shù)的導(dǎo)數(shù)仍為初等函數(shù).6/20/2023回顧：變速直線運(yùn)動(dòng)的加速度.定義1.定義八、高階導(dǎo)數(shù)6/20/2023記作三階導(dǎo)數(shù)的導(dǎo)數(shù)稱為四階導(dǎo)數(shù),二階和二階以上的導(dǎo)數(shù)統(tǒng)稱為高階導(dǎo)數(shù).二階導(dǎo)數(shù)的導(dǎo)數(shù)稱為三階導(dǎo)數(shù),6/20/20232.高階導(dǎo)數(shù)求法舉例例1解直接法:由高階導(dǎo)數(shù)的定義逐步求高階導(dǎo)數(shù).6/20/2023例2解注意:

求n階導(dǎo)數(shù)時(shí),求出