導(dǎo)數(shù)與微分

1、導(dǎo)數(shù)與微分,一、導(dǎo)數(shù)的概念1.自變量的增量：2.函數(shù)的增量： 3.導(dǎo)數(shù)的定義：,,,,導(dǎo)數(shù)與微分,即導(dǎo)數(shù)為函數(shù)增量與自變量增量比的極限,,,,導(dǎo)數(shù)與微分,,,,導(dǎo)數(shù)與微分,二、導(dǎo)數(shù)的物理和幾何意義1.物理意義： 表示運動物體瞬時速度即：2.幾何意義： 表示曲線y＝f(x)在x0處的切線斜率即 若切點為 則曲線在 的 切線方程為： 法線

2、方程為：,,,,,,,,,,,,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,三、基本求導(dǎo)公式：,,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,四、求導(dǎo)法則若u=u(x)，v=v(x)在x處可導(dǎo)，則,,導(dǎo)數(shù)與微分,1.求下列函數(shù)的導(dǎo)數(shù),,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,2.復(fù)合函數(shù)求導(dǎo),,導(dǎo)數(shù)與微分,注：復(fù)合函數(shù)求導(dǎo)法則的關(guān)鍵在于：（1) 將復(fù)合函數(shù)分解成若干個基本初等函數(shù)；（2) 分別求出這些函數(shù)的導(dǎo)數(shù)并相乘；（3) 將所設(shè)中間變

3、量還原,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,例5：證明：偶函數(shù)的導(dǎo)數(shù)是奇函數(shù)。證：設(shè)f(x)是偶函數(shù)，即f(-x)=f(x)u=-x,,導(dǎo)數(shù)與微分,3.隱函數(shù)求導(dǎo)法則：隱函數(shù)：由含x，y的方程F(x,y）＝0給出的函數(shù)稱為隱函數(shù)。有些方程，可以從中解出y，將y表示成x的顯函數(shù)的形式。如：有些方程則不能解出y，如

4、 等，對于這樣的隱函數(shù)可不必解出y，而是將y作為x的函數(shù)隱藏在方程中利用隱函數(shù)求導(dǎo)法則求出其導(dǎo)數(shù),,,導(dǎo)數(shù)與微分,隱函數(shù)的求導(dǎo)法則：將y作為x的函數(shù)，y＝y(tǒng)(x)，于是F(x，y(x)）＝0對方程兩邊的x求導(dǎo)，遇y時，將y作為中間變量，利用復(fù)合函數(shù)求導(dǎo)法則對y求導(dǎo)再乘 得到一個含的方程，最后從新方程中解出,,,,,,,,,,,,導(dǎo)數(shù)與微分,例6：求下列函數(shù)的導(dǎo)數(shù),,導(dǎo)數(shù)與微分,,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)

5、與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,注：對一些較復(fù)雜的乘積，商或根式函數(shù)求導(dǎo)時，可利用先取對數(shù)后求導(dǎo)的方法計算,,導(dǎo)數(shù)與微分,5.參數(shù)方程求導(dǎo)法則,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,五、函數(shù)的微分1.微分的定義:設(shè)函數(shù)y=f(x)在點x0處可導(dǎo), 是自變量x的增量,則稱 為函數(shù)f(x)在x0處關(guān)于?x的微分.記為: ,即2.函數(shù)可微的條件: 定理: 函數(shù)y=f

6、(x)在x點可微的充分必要條件是y=f(x)在x點處可導(dǎo). 即：函數(shù)可微 存在,則函數(shù)可導(dǎo)且 ,反之,函數(shù)可導(dǎo),既 存在,則 從而函數(shù)可微.,,,,,,,,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,3.微分公式,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,4.微分法則,,導(dǎo)數(shù)與微分,例10 求下列函數(shù)的微分：,,導(dǎo)數(shù)與微

7、分,,導(dǎo)數(shù)與微分,5.一階微分形式不變性：若u為自變量,y＝f(u),則,若u為中間變量,從而不論u是自變量還是中間變量其微分的形式不變,皆為dy=f’(x)du.我們將微分的這一性質(zhì)稱為一階微分形式不變性利用一階微分形式不變性可以方便的求出復(fù)合函數(shù)和隱函數(shù)的微分和導(dǎo)數(shù)。,,,導(dǎo)數(shù)與微分,,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,例12 求下列隱函數(shù)的微分和導(dǎo)數(shù),,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,,導(dǎo)數(shù)與微分,6.微分在近