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思考與總結(jié)

一、導(dǎo)數(shù)與微分的概念與結(jié)論

1.導(dǎo)數(shù)的概念

(1)導(dǎo)數(shù)的定義:__________.

(2)導(dǎo)數(shù)f′(x0)的幾何意義:__________,切點(diǎn)處的切線方程:__________;法線方程:__________.

(3)導(dǎo)數(shù)f′(t0)的物理意義:__________;二階導(dǎo)數(shù)f″(t0)的物理意義:__________.

(4)函數(shù)在x0點(diǎn)連續(xù)與可導(dǎo)的關(guān)系:__________.

(5)高階導(dǎo)數(shù)的記號與求法:__________.

2.微分的概念

(1)f(x)在任意點(diǎn)x處的微分:__________.

(2)微分與導(dǎo)數(shù)的關(guān)系:__________.

二、導(dǎo)數(shù)與微分的基本公式與運(yùn)算法則

1.基本初等函數(shù)的導(dǎo)數(shù)公式

(1)(c)′=__________(c為常數(shù));  (2)(xα)′=__________(α為任意實(shí)數(shù));

(3)(x)′=__________;  (4)(x2)′=__________;

(7)(ax)′=__________(a>0,a≠1);(8)(ex)′=__________;

(9)(logax)′=__________; (10)(ln x)′=__________;

(11)(sin x)′=__________; (12)(cos x)′=__________;

(13)(tan x)′=__________; (14)(cotx)′=__________;

(15)(secx)′=__________; (16)(cscx)′=__________;

(17)(arcsin x)′=__________; (18)(arccos x)′=__________;

(19)(arctan x)′=__________; (20)(arccotx)′=__________.

2.導(dǎo)數(shù)的四則運(yùn)算法則

(1)(u±v)′=__________;  (2)(uv)′=__________;

(3)(cu)′=__________;  (4)=__________(v≠0);

(5)=__________(v≠0,c為常數(shù))

3.復(fù)合函數(shù)的求導(dǎo)法則:__________.

(1)若u=u(x)可導(dǎo),則(sin u)′x=__________;(ln u)′x=__________.

(2)若u=2x,則(cos 2x)′=__________;(arctan 2x)′=__________.

4.隱函數(shù)的求導(dǎo)法:__________.

5.微分形式的不變性:__________.

(1)d(eu)=__________;(2)(uα)′=__________(α∈R);(3)(tan u)′=__________.

三、可導(dǎo)、可微、連續(xù)與極限存在的關(guān)系:

可微__________可導(dǎo)__________連續(xù)__________極限存在.

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