2-2导数求导法则ppt

1.3高阶导数的定义
问题:变速直线运动的加速度.
设 s = s (t ), 则瞬时速度为 v(t ) = s' (t )
加速度a是速度v对时间 t的变化率
∴ a (t ) = v′(t ) = [ s ' (t )]′.
定义 如果函数f ( x )的导数f ′( x )在点x处可导, 即 f ′( x + ∆ x ) − f ′( x ) ( f ′( x ))′ = lim ∆x → 0 ∆x 存在, 则称( f ′( x ))′为函数f ( x )在点x处的二阶导数.
y ( n ) = ( x n )( n ) = n! ,
y ( n + 1) = ( n! )′ = 0.
预习1.4 微分;来自d3y . 二阶导数的导数称为三阶导数, f ′′′( x ), y′′′, 3 dx 4 d y (4) (4) . 三阶导数的导数称为四阶导数, f ( x ), y , 4 dx
d 2 y d 2 f ( x) . 记作 f ′′( x ), y ′′, 2 或 2 dx dx
一般地, 函数f ( x )的n − 1阶导数的导数称为
x可导, 且其导数为 dy dy du = ⋅ . dx du dx
即 因变量对自变量求导,等于因变量对中间变 量求导,乘以中间变量对自变量求导.(链式法则)
例5 求函数 y = ln sin x 的导数 . 解
y = ln u, u = sin x . 1 dy dy du 1 ⋅ cos x = ⋅ cos x = ∴ = ⋅ sin x dx du dx u cos x = = cot x sin x
y ′′ = (αx α − 1 )′ = α(α − 1) x α − 2 y ′′′ = (α(α − 1) x α − 2 )′ = α(α − 1)(α − 2) x α − 3

y ( n ) = α(α − 1) (α − n + 1) x α − n ( n ≥ 1)
若 α = n, 则
(cot x)′ = − csc 2 x (csc x)′ = − csc x cot x
(a )′ = a ln a 1 (log a x)′ = x ln a
x x
(e x )′ = e x 1 (ln x)′ = x
四则运算公式
如果函数 u = u ( x), v = v( x)在点 x处可导, 则有
可导与连续的关系
定理 凡可导函数都是连续函数. 证 要证 lim f ( x) = f ( x0 ) 只需证 lim f(x)-f(x0 ) = 0
x → x0 x-x0 →0
即 lim ∆y = 0
∆x →0
设函数 f ( x )在点 x0可导,
∆y = f ′( x 0 ) lim ∆x → 0 ∆ x
( xα )′ = αxα −1 (cos x)′ = − sin x
(cot x)′ = − csc 2 x (csc x)′ = − csc x cot x
(a )′ = a ln a 1 (log a x)′ = x ln a
x x
(e x )′ = e x 1 (ln x)′ = x
y' 1.设y = x sin x, 求y '. =
§1 导数 1.1 导数定义 1.2 求导法则 1.3 高阶导数
1.四则运算
如果函数 u = u ( x), v = v( x)在点 x处可导, 则有
(1) [u ± v]′ = u ′ ± v′;
(2) [u ⋅ v]′ = u ′ ⋅ v + u ⋅ v′;
u u ′ ⋅ v − u ⋅ v′ (3) [ ]′ = (v ≠ 0). 2 v v
x2 +1 2.求函数 y = ln 的导数. x
−1 y'= . 2 x( x + 1)
初等函数的求导
常数和基本初等函数的导数公式
(c)′ = 0 (sin x)′ = cos x (tan x)′ = sec x (sec x)′ = sec x tan x
2
( xα )′ = αxα −1 (cos x)′ = − sin x
= sec x tan x .
(sec x)′ = sec x tan x.
同理可得
(csc x )′ = − csc x cot x .
常数和基本初等函数的导数公式
(c)′ = 0 (sin x)′ = cos x (tan x)′ = sec x (sec x)′ = sec x tan x
2
1 =− 2e x
sin 1 x
sin
1 x
dy dy du dv 1 u = ⋅ ⋅ = e ⋅ cos v ⋅ (− 2 ) dx du dv dx x
1 ⋅ cos . x
1.设y = ( x 2 + 1) x , 求y '.
2 x 2 y ' = ( x 2 + 1) x [ln( x 2 + 1) + 2 ] x +1
推论
C
(1) [∑ f i ( x )]′ = ∑ f i′( x );
i =1 i =1
n
n
( 2) [Cf ( x )]′ = Cf ′( x );
(3) [∏
i =1 n
′ ′ f i ( x)] = f1 ( x) f 2 ( x) f n ( x) + + f1 ( x) f 2 ( x) f n′( x)
推广 设 y = f ( u), u = ϕ ( v ), v = ψ ( x ),
则复合函数 y = f {ϕ [ψ ( x )]}的导数为 dy dy du dv = ⋅ ⋅ . dx du dv dx
例7 解
求函数 y = e 的导数 . 1 u y = e , u = sin v, v = x
(1) [u ± v]′ = u ′ ± v′; (2) [u ⋅ v]′ = u ′ ⋅ v + u ⋅ v′; u u ′ ⋅ v − u ⋅ v′ (3) [ ]′ = (v ≠ 0). 2 v v
复合函数求导法则
dy dy du = ⋅ . dx du dx
关键: 正确分解初等函数的复合结构.
函数 f ( x) = x 在x = 0处连续但不可导.
小结
1. 导数的实质: 增量比的极限; 2. 求导数最基本的方法: 由定义求导数; 3. 导数的几何意义和物理意义; 4. 函数可导一定连续，但连续不一定可导;
不连续,一定不可导.
5. 判断可导性
连续
直接用定义; 看左右导数是否存在且相等.
第三章 导数及其应用
u ( x + h )v ( x ) − u ( x )v ( x ) + u ( x )v ( x ) − u ( x )v ( x + h ) = lim h →0 v ( x + h )v ( x ) h
[u( x + h) − u( x )]v ( x ) − u( x )[v ( x + h) − v ( x )] = lim h→ 0 v ( x + h)v ( x )h
u( x + h) − u( x ) v ( x + h) − v ( x ) ⋅ v ( x ) − u( x ) ⋅ h h = lim h→ 0 v ( x + h)v ( x )
u′( x )v ( x ) − u( x )v ′( x ) = [v ( x )]2
∴ f ( x )在x处可导.
∆y α= − f ′( x0 ) ∆x
∆ y = f ′( x 0 ) ∆ x + α ∆ x
α → 0 ( ∆x → 0 )
∆x → 0 ∆x → 0
lim ∆y = lim [ f ′( x 0 )∆x + α∆x ] = 0
∴ 函数 f ( x )在点 x 0 连续 .
注意: 该定理的逆命题不成立.
= 2 cos x ⋅ cos x ⋅ ln x + 2 sin x ⋅ ( − sin x ) ⋅ ln x 1 + 2 sin x ⋅ cos x ⋅ x 1 = 2 cos 2 x ln x + sin 2 x . x
y′ = (2 sin x)'⋅ cos x ⋅ ln x + 2 sin x ⋅ (cos x)'⋅ ln x + 2 sin x ⋅ cos x ⋅ (ln x)'
2.求函数 y =
2
x2 +1 的导数. x
y' =
x2 x2 + 1
dy 3.设y = x ln( x + 1), 求 . dx 4.设y = sin 2 x dy ,求 . x dx
2 dy 2 x = ln( x 2 + 1) + 2 dx x +1
dy 2 x cos 2 x − sin 2 x = dx x2
C
u u ′ ⋅ v − u ⋅ v′ [ ]′ = (v ≠ 0). 2 v v 例3 求 y = tan x 的导数 .
解
sin x )′ y ′ = (tan x )′ = ( cos x
(sin x )′ cos x − sin x (cos x )′ = cos 2 x
1 cos 2 x + sin 2 x 2 = = sec x = 2 2 cos x cos x
u( x ) 证(3) 设 f ( x ) = , (v ( x ) ≠ 0), v( x )
f ( x + h) − f ( x ) f ′( x ) = lim h→ 0 h u( x + h) u( x ) − v ( x + h) v ( x ) = lim h→ 0 h