考研數(shù)學(xué)第二章導(dǎo)數(shù)與微分

2023/5/151第二章導(dǎo)數(shù)與微分教學(xué)要求：1.導(dǎo)數(shù)定義的理解與可導(dǎo)性的判斷；可導(dǎo)性與連續(xù)性的關(guān)系；分段函數(shù)的可導(dǎo)性與導(dǎo)數(shù)；利用導(dǎo)數(shù)定義求極限，2.求平面曲線的切線方程與法線方程；3.微分定義的理解；函數(shù)的增量、微分、自變量的增量與自變量的增量與導(dǎo)數(shù)的關(guān)系；4.求積分上限函數(shù)的導(dǎo)數(shù)；5.利用四則運(yùn)算求導(dǎo)數(shù)；2023/5/1526.求復(fù)合函數(shù)的導(dǎo)數(shù)；7.求反函數(shù)的導(dǎo)數(shù)；8.求隱函數(shù)的導(dǎo)數(shù)，對(duì)數(shù)求導(dǎo)法；9.求參數(shù)方程的導(dǎo)數(shù)；10.求高階導(dǎo)數(shù)；2023/5/153一、導(dǎo)數(shù)的概念定義1.

設(shè)函數(shù)在點(diǎn)存在,并稱此極限為記作:即則稱函數(shù)若自變量的某鄰域內(nèi)有定義,在點(diǎn)處可導(dǎo),在點(diǎn)的導(dǎo)數(shù).在處取得增量時(shí)，相應(yīng)地函數(shù)取得增量若極限2023/5/154導(dǎo)數(shù)定義的幾種形式：2023/5/155右導(dǎo)數(shù):左導(dǎo)數(shù):單側(cè)導(dǎo)數(shù)：導(dǎo)函數(shù)：2023/5/156例1√2023/5/157例22023/5/158例32023/5/159解例42023/5/1510解2023/5/1511解2023/5/1512解2023/5/1513解分段函數(shù)的可導(dǎo)性與導(dǎo)數(shù)例52023/5/1514[分析]例62023/5/1515而事實(shí)上，2023/5/1516[分析]求分段函數(shù)在分段點(diǎn)的導(dǎo)數(shù)要用定義2023/5/1517[分析]例72023/5/1518[分析]2023/5/1519[分析]2023/5/15202023/5/1521[分析]2023/5/15222023/5/1523利用導(dǎo)數(shù)定義求極限2023/5/1524二、求曲線的切線和法線方程2023/5/15252023/5/15262023/5/15272023/5/152852023/5/1529的微分,定義:

若函數(shù)在點(diǎn)的增量可表示為(A

為不依賴于△x

的常數(shù))則稱函數(shù)而稱為記作即定理:

函數(shù)在點(diǎn)可微的充要條件是即在點(diǎn)可微,三、微分，微分與導(dǎo)數(shù)的關(guān)系2023/5/1530微分的幾何意義MNT）

P2023/5/15312023/5/15322023/5/15332023/5/15342023/5/15352023/5/1536四、變限積分求導(dǎo)2023/5/15372023/5/15382023/5/15392023/5/15402023/5/15412023/5/1542五、利用四則運(yùn)算求導(dǎo)2023/5/15432023/5/1544七、

復(fù)合函數(shù)的求導(dǎo)法則(鏈?zhǔn)椒▌t)六、

反函數(shù)的導(dǎo)數(shù)2023/5/15452023/5/15462023/5/15472023/5/15482023/5/15492023/5/15502023/5/15512023/5/1552解2023/5/15532023/5/15542023/5/1555八、隱函數(shù)的導(dǎo)數(shù)隱函數(shù)求導(dǎo)法則:用復(fù)合函數(shù)求導(dǎo)法則直接對(duì)方程兩邊求導(dǎo).2023/5/15562023/5/1557[解]所求法線方程為2023/5/1558解2023/5/1559解2023/5/1560對(duì)數(shù)求導(dǎo)法2023/5/15612023/5/1562[例][解]對(duì)數(shù)求導(dǎo)法2023/5/15632023/5/1564九、由參數(shù)方程所確定的函數(shù)的導(dǎo)數(shù)2023/5/15652023/5/15662023/5/15672023/5/156