高数2-2

1 1′ ⋅ cos x − 1 ⋅ (cos x )′ y ′ = (sec x )′ = ( )′ = cos x cos 2 x
− (cos x )′ sin x = sec x tan x . = = 2 2 cos x cos x
即
同理可得
(sec x )′ = sec x tan x .
u( x + h)v ( x ) − u( x )v ( x + h) = lim h→ 0 v ( x + h)v ( x )h
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
证(1)、(2)略.
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
即
(arcsin x )′ =
1 1− x
2
.
同理可得 (arccos x )′ = −
1 (arctan x )′ = ; 2 1+ x
1 1− x
2
.
1 (arc cot x )′ = − . 2 1+ x
例6
求函数 y = log a x 的导数 .
解 ∵ x = a y在I y ∈ ( −∞ ,+∞ )内单调、可导 ,
切记： f ′( x 2 ) = f ′( u ) [ f ( x ) ]′ = f ′( u )
2 u= x 2
u= x 2 2
;
2 ′ ′ ⋅ ( x ) = 2 xf ( x )
四、基本求导法则与求导公式
1.常数和基本初等函数的导数公式
( C )′ = 0
μ −1 ( x μ )′ = μx
dy dy du 1 cos x = cot x ∴ = ⋅ = ⋅ cos x = dx du dx u sin x
例8 解
求 y = shx 的导数 .
1 x 1 x −x −x y′ = ( shx )′ = [ (e − e )]′ = (e + e ) = chx . 2 2
( shx )′ = chx .
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
Байду номын сангаас
例7 求函数 y = ln sin x 的导数 . 解
∵ y = ln u, u = sin x .
例5 求函数 y = arcsin x 的导数 .
π π 解 ∵ x = sin y在 I y ∈ ( − , )内单调、可导 , 2 2 且 (sin y )′ = cos y > 0, ∴ 在 I x ∈ ( −1,1)内有
1 1 1 1 = = (arcsin x )′ = = . 2 2 (sin y )′ cos y 1 − sin y 1− x
x0可导, 且其导数为 dy dx
x = x0
= f ′( u0 ) ⋅ ϕ′( x0 ).
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)
证
Δy = f ′( u0 ) 由y = f ( u)在点 u0可导 , ∴ lim Δu → 0 Δ u Δy 故 = f ′( u0 ) + α ( lim α = 0) Δu→ 0 Δu
= lim [u( x + h) − u( x )]v ( x ) − u( x )[v ( x + h) − v ( x )] 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
证 任取 x ∈ I x , 给x以增量 Δx ( Δx ≠ 0, x + Δx ∈ I x )
由y = f ( x )的单调性可知
于是有
Δ y ≠ 0,
Δy 1 , = Δx Δx Δy
∵ f ( x )连续 ,
∴ Δy → 0 ( Δx → 0),
又知 ϕ ′( y ) ≠ 0
Δy 1 1 ∴ f ′( x ) = lim = = lim Δx → 0 Δ x Δy → 0 Δ x ϕ ′( y ) Δy 1 即 f ′( x ) = . ϕ ′( y )
(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
即
同理可得
(tan x )′ = sec 2 x .
(cot x )′ = − csc2 x.
例4 求 y = sec x 的导数 . 解
(csc x )′ = − csc x cot x
(arcsin x )′ =
(arctan x )′ =
1 1− x
2
(arccos x )′ = −
1 1 − x2
1 1 + x2
1 (arc cot x )′ = − 1 + x2
2.函数的和、差、积、商的求导法则 设 u = u( x ), v = v ( x ) 可导，则 , （2）(cu)′ = cu′ ( C 是常数) （1）( u ± v )′ = u′ ± v ′
x2 + 1 例11 求函数 y = ln 3 ( x > 2) 的导数 . x−2
1 1 2 解 ∵ y = ln( x + 1) − ln( x − 2), 2 3 1 1 1 1 x ⋅ 2x − = 2 − ∴ y′ = ⋅ 2 2 x +1 3( x − 2) x + 1 3( x − 2)
2
2
a
2 2 1 x a 1 2 + = a − x2 − 2 2 2 a −x 2 a2 − x2 2
= a2 − x2 .
( a − x )′ =
2 2
(a − x )′
2 2
2 a −x
2
2
=
−x a −x
2 2
x (arcsin )′ = a
x ( )′ a = x 1 − ( )2 a
1 a2 − x2
则 Δy = f ′( u0 )Δu + αΔu
Δy Δu Δu ∴ lim +α ] = lim [ f ′( u0 ) Δx → 0 Δ x Δx → 0 Δx Δx Δu Δu ′ + lim α lim = f ( u0 ) lim ′( u0 )ϕ ′( x 0 ). = f Δx → 0 Δ x Δx → 0 Δx → 0 Δ x
(csc x )′ = − csc x cot x .
二、反函数的导数
定理 如果函数 x = ϕ ( y )在某区间 I y内单调、可导 且ϕ ′( y ) ≠ 0 , 那末它的反函数 y = f ( x )在对应区间 I x内也可导 , 且有
1 . f ′( x ) = ϕ ′( y )
即 反函数的导数等于直接函数导数的倒数.
fn ( x)
+
+ f1 ( x ) f 2 ( x )
例1 求 y = x − 2 x + sin x 的导数 .
3 2
解
y ′ = 3 x 2 − 4 x + cos x .
例2 求 y = sin 2 x ⋅ ln x 的导数 . ∵ y = 2 sin x ⋅ cos x ⋅ ln x 解
(a x )′ = a x ln a
1 (log a x )′ = x ln a
(e x )′ = e x
1 (ln x )′ = x
(sin x )′ = cos x
(tan x )′ = sec x
2
(cos x )′ = − sin x
2 (cot x )′ = − csc x
(sec x )′ = sec x tan x
′ v − uv ′ u u ′ ′ ′ ( v ≠ 0) . （3）( uv ) = u v + uv , （4）( )′ = 2 v v
3.复合函数的求导法则
设y = f ( u), 而u = ϕ ( x )则复合函数 y = f [ϕ ( x )]的 dy dy du 导数为 = ⋅ 或 y′( x ) = f ′( u) ⋅ ϕ ′( x ). dx du dx
∴ f ( x )在x处可导.
推论
(1) [∑ f i ( x )]′ = ∑ f i′( x );
i =1 i =1 n n
( 2) [Cf ( x )]′ = Cf ′( x );
( 3)[∏ f i ( x )]′ = f1′( x ) f 2 ( x )