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www54549911com:函数的和、差、积、商的导数二_图文

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知识回顾:

基本求导公式:

(1)C? ? 0(C为常数)
(?为常数)

(2)( x ) ? ?x
x ' x
'

? '

? ?1

(3)(a ) ? a lna(a ? 0, 且a ? 1)
1 (4)(log a x ) ? (a ? 0, 且a ? 1) xlna 1 ' x ' x (6)(lnx) ? (5)(e ) ? e x

(7)(sinx ) ? cosx
'

(8)(cosx) ? ?sinx
'

练习:
1.求函数下列函数的导数.

(1) f ( x) ? x

2

(2) g ( x) ? x

(3)h( x) ? x ? x
2

2.从以上计算结果,你能发现什么?

[ f ( x) ? g ( x)] ? f ( x) ? g ( x)
' ' '

两个函数的和的导数,等于这两个函数导数的和. 以上两个函数的差的导数,等于这两个函数导数的差吗?

函数的和、差、积、商 的导数

证明猜想
证明:令

? f ( x) ? g ( x)?
y ? f ( x) ? g ( x).

?

? f ?( x) ? g ?( x).

?y ? ? f ( x ? ?x) ? g ( x ? ?x)?? ? f ( x) ? g ( x)?

? ? f ( x ? ?x) ? f ( x)? ? ?g ( x ? ?x) ? g ( x)?
?y ? f ( x ? ?x) ? f ( x)? ? ?g ( x ? ?x) ? g ( x)? ? ? ?x ?x

? f ( x ? ?x) ? f ( x)? ?g ( x ? ?x) ? g ( x)? ? ?
?x ?x

? f ?( x) ? g ?( x)

法则1: 两个函数的和(或差)的 导数,等于这两个函数的导数的和 (或差),即:

[ f ( x) ? g ( x)]? ? f ?( x) ? g ?( x).
法则2:

[Cf ( x)]? ? Cf ?( x).(C为常数)
说明:求导法则的证明不作要求.

例1. (1)求函数f ( x) ? x ? sin x的导数.
2

解:f ?( x) ? ( x ? sin x)?
2

? ( x )? ? (sin x)? ? 2 x ? cos x
2

3 2 (2)求函数g ( x) ? x ? x ? 6 x ? 2的导数. 2
3

3 2 解:g ?( x) ? ( x ? x ? 6 x ? 2)? 2 3 2 3 ? ( x )? ? ( x )? ? (6 x)? ? 2' ? 3 x 2 ? 3 x ? 6 2
3

法则 3: 两个函数的积的导数,等于

第一个函数的导数乘以第二个函数 ,
加上第一个函数乘以第二个函数的导 数

[ f ( x) g ( x)]? ? f ?( x) g ( x) ? f ( x) g ?( x).

例2: (1)求函数h( x) ? x sin x的导数. (2)求函数f ( x) ? 2 x ln x的导数.
解 : (1)h?( x) ? ( x sin x)? ? x? sin x ? x(sin x)? ? sin x ? x cos x

(2) f ?( x) ? (2 x ln x)? ? (2 x)? ln x ? (2 x)(ln x)? ? 2 ln x ? 2

?3?.用两种方法求y ?(2x
的导数

2

? 3)(3x? 2)

2 2 解: 法一:y? ? (2x ? 3)?(3x ? 2) ? (2x ? 3)(3x ? 2)?

? 4 x ( 3 x ? 2) ? ( 2 x ? 3) ? 3
2

? 18 x ? 8 x ? 9 3 2 法二: y? ? (6 x ? 4 x ? 9 x ? 6)?
2

? 18 x ? 8 x ? 9
2

法则4 :两个函数的商的导数,等于分

子的导数与分母的积,减去分母的导数 与分子的积,再除以分母的平方,即:

f ( x) f ?( x) g ( x) ? f ( x) g ?( x) [ ]? ? 2 g ( x) g ( x)

其中g ( x) ? 0

t ?1 例3: (1)求函数s (t ) ? t 的导数.
2

( 2)求函数y ? t an x的导数

x (4)求函数f(x) ? x 的导数. e

cos x (3 )求函数y ? 的导数 x

x (4)求函数f(x) ? x 的导数. e x
e x x x x x?e ? x(e )? e ? xe 1 ? x ? ? ? x 2 2x x (e ) e e
课本P22 练习

f ?( x) ? (

? ) x

函数的求导法则:
(1)[ f ( x) ? g ( x)] ? f ( x) ? g ( x)
' ' '

(2)[ f ( x) ? g ( x)]' ? f ' ( x) ? g ' ( x) (3)[Cf ( x)]' ? Cf ' ( x)(C为常数) (4)[ f ( x) g ( x)]' ? f ' ( x) g ( x) ? f ( x) g ' ( x)
f ( x) ' f ( x) g ( x) ? f ( x) g ( x) (5)[ ]? 2 g ( x) g ( x)
' '

练习:
1.求下列函数的导数:
(1) y ? (2 x ?1)(3x ? 2)
x x (2) y ? 1 ? sin cos 2 2 x (4) y ? 2x ? 3

(3) y ? 2x ? 2ln x
2.求下列函数的导数:
(1) y ? (tan x).x
1 (3) y ? e ? 2 x
?x

(2) y ? x sin x ? x ln x (4) y ? ? 2x x ?1

思考讨论:
1.(1)已知函数f(x)的导数为 f ' ( x),则[ f ( x)]2 的导数 为 2 f ' ( x) ,这句话对吗? (2)已知函数f(x)的导数为 3 x 2 ,则函数f(x) ? x3 . 2.(1)在曲线y=x3+x-2上求一点P,使曲线在P处的切 线平行于直线4x-y-7=0. (2)已知直线x-2y+4=0和抛物线x2=4y相交于A,B 两点,在弧AOB上求一点P,使△APB的面积最大,求 点P的坐标.

(2)已知直线x-2y+4=0和抛物线x2=4y相交于A,B 两点,在弧AOB上求一点P,使△APB的面积最大,求 点P的坐标.
y
B

A

P

O

x


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