1
第二节 导数的运算法则
习题 2-2
1,推导余切函数及余割函数的导数公式,
(1)
2
(cot ) cscx x′=? ; (2) (csc ) csc cotx xx′=?,
解 (1)
2
22
cos (cos ) sin (sin ) cos 1
(cot ) ( ) csc
sin sin sin
xxxxx
x x
x xx
′ ′?
′′== =?=?,
(2)
2
11
(csc ) ( ) (sin ) csc cot
sin sin
x xxx
x x
′′ ′==? =?,
2,求下列函数的导数,
(1)
3
2
3
27yx
x
=?+; (2) ln 2 2
x
yx x= ++;
(3) 2csc cotyxx=+; (4) e arccos
x
yx= ;
(5)
3
2
log
x
yx= ; (6)
ln x
y
x
= ;
(7)
2
ln cosyx x x= ; (8)
2
2
1
1
x
y
x
+
=;
(9) ( 0)
ax
yxa a=>; (10)
1sin
1cos
t
s
t
+
=
+
,
解 (1)
32
23
36
(2 ) ( ) (7) 6yx x
x x
′′′′=?+=+,
(2)
(2 ) 1
(ln2 ) (2 ) ( ) 2 ln2 1 2 ln2 1
2
xxx
x
yx x
xx
′
′′′′=++=++=++,
(3)
2
(2csc ) (cot ) 2csc cot cscyxx xxx′′′=+=,
(4) (e ) arccos e (arccos )
xx
yxx′′ ′=+
22
11
e arccos e e (arccos )
xxx
x x
=?=?
,
(5)
33 232
22 2 2
11
( ) log (log ) 3 log (3log )
ln 2 ln 2
xx x x
yx x x x x
x
′′ ′=+=+=+,
2
(6)
22
(ln ) ln 1 lnx xx x
y
′
′==,
(7)
22 2
( ) ln cos (ln ) cos ln (cos )yx xxxx xxx x′′ ′ ′=++
2
2ln cos cos ln sinx xxxxxxx=+?,
(8)
22 22
22 22
(1 ) (1 ) (1 )(1 ) 4
(1 ) (1 )
x xxx x
y
′′++?
′==
,
(9)
11
() () ln (ln )
ax ax ax ax ax
yxaxa axax axaxaa
′′ ′=+=+ = +,
(10)
22
(1 sin ) (1 cos ) (1 sin )(1 cos ) 1 cos sin
(1 cos ) (1 cos )
tt tt tt
s
tt
′′++?++ ++
′==
++
,
3,以初速
0
v 竖直上抛的物体,其上升高度 s 与时间 t 的关系是
2
0
1
2
s vt gt=?,
求,
(1) 该物体的速度 ()vt ;
(2) 该物体到达最高点的时刻,
解 (1)
0
()vt s v gt′==?,
(2) 物体到达最高点时,( ) 0vt =,即
0
0vgt? =,从而
0
v
t
g
=,
4,求曲线 (ln 1)yx x=?上横坐标为 ex = 的点处的切线方程和法线方程,
解 该点为 (e,0),所求切线的斜率为
eee
(ln 1 1) ln 1
xxx
yx
===
′ =?+ = =,从而切线方程为,eyx=?,法线方程为,eyx=?+,
5,求下列函数的导数,
(1)
2
3
e
x
y
= ; (2)
2
cos(4 3 )yx=?;
(3) arctan(e )
x
y = ; (4)
2
(arcsin )yx= ;
(5)
2
tan
(0)
x
ya a=>; (6)
23
cos (tan )yx= ;
(7)
2
1
sin
2
x
y = ; (8)
x
yx=,
解 (1)
22
32 3
e(3) 6e
x x
yx
′′=?=?,
3
(2)
22 2
sin(43)(43) 6sin(43)yxxxx′′= =?,
(3)
22
(e ) e
1e 1e
xx
x x
y
′
′==
++
,
(4)
2
2arcsin
2(arcsin )(arcsin )
1
x
yxx
x
′′==
,
(5)
22 2
tan 2 tan 2 2 2 2 2 tan
ln (tan ) ln sec ( ) 2ln sec
x xx
ya a x a a xx ax xa′′ ′== =?,
(6)
33 333
2cos(tan )[cos(tan )] 2cos(tan )sin(tan )(tan )yxx xxx′?
32
3sin(2 tan ) tan (tan )x xx′=?
22 3
3tan sec sin(2 tan )x xx=?,
(7)
22
11
sin sin
2
111
2 ln2(sin ) 2 2ln2sin (sin )
xx
y
x xx
′′ ′==
22
sin sin
2
111 ln2 2
22ln2sincos() sin
xx
x xx xx
′==?,
(8)
11
ln ln
222
1111ln
(e ) e ( ln ) ( ln )
xx
x
yxxx
x xxx
′′ ′== =?+=,
6,求下列函数在指定点处的导数值,
(1)
2
() sin3,0
1
f
= +=;
(2)
13
arcsin 2,
4
yxx
x
==;
(3)
5
3e 5(1 ),1
x
yxx
==?,
解 (1)
22 2
22 22
12 1
() 3cos3 3cos3,(0) 4
(1 ) (1 )
ff
+ +
′′=+ =+ =
,
(2) 3
2
2
4
1216
arcsin 2,(3 3 π )
9
14
x
yxy
x
xx
=
′′=? + =?
,
(3)
55
1
15e 5,5(1 3e )
x
x
yy
=?
′′=? + =?,
4
7,求下列函数的导数,
(1) ln(sec tan )yxx=+; (2)
1
ln tan arctan( tan )
222
x x
y =+ ;
(3) sin cos
n
yxnx= ; (4)
1
arcsin
1
x
y
x
=
+;
(5)
2
2
sin
sin
x
y
x
= ; (6)
2
1
1
y
x x
=
++;
(7)
2
arcsin 4
2
x
yx x=+?; (8)
2
sh ch3yx
x
= ;
(9)
2
1
ln ch
2ch
yx
x
=+ ; (10)
xaa
aax
ya x a=++,
解 (1)
2
sec tan sec
sec
sec tan
xx x
xx
+
′==
+
,
(2)
22
22
11
sec sec
1
2242
csc
1
tan 1 tan 1 3cos
242 2
xx
yx
x xx
′=+ =+
++
,
(3)
11
sin cos sin sin sin cos( 1)
nnn
y n xcox nx n x nx n x n x
′=?=+,
(4)
2
11(1)(1) 1
(1 )11 (1)2(1)
12
11
xx
y
xx x
xx
+
′==
+ +?
++
,
(5)
22 2 2 2
43
2 cos sin sin 2sin cos 2 cos sin 2sin cos
sin sin
x xxxxxxxx xx
y′,
(6)
22 2 2 2
12 1
(1 )
(1) 21 1(1)
x
y
x xxxx
′=? + =?
++ + + ++
,
(7)
22
11 2
arcsin arcsin
22 2
24
1
4
x xx
yx
xx
′=+ + =
,
(8)
22
22 2 22 2
ch ( )ch3 3sh sh3 ch ch3 3sh sh3yxxxx
xx
′=? + =? +,
(9)
3
32
sh sh 1
2th(1)th
ch 2ch ch
xx
yxx
x xx
′=? =? =,
(10)
1
ln ( ) ln ( )
xaa
axa xa
ya aa x a ax
′′ ′=++
5
211
ln ln
xa a
x aaa ax
aa a a x a ax a
=++,
8,设 ()f x 和 ()g x 都可导,求下列函数 y 的导数
d
d
y
x
,
(1)
()
(e )e
x fx
yf= ; (2)
22
(sin ) (cos )yf x f x=+;
(3)
2
ln ( ) arctan ( )yfx gx=+ ; (4)
2
() ()yfxgx=+,
解 (1)
() () () ()
(e)ee (e)e () (e)e (e)e ()
x x fx x fx x fx x x fx
yf ffxf ffx
+
′ ′′′′=+ =,
(2)
22
(sin )2sin cos (cos )2sin cosyf xxxf xxx′′ ′=?
22
sin 2 [ (sin ) (cos )]x fxfx′′=?,
(3)
22 22
()1 ()2 () 2()
1() 1()()2 2 ()
f xgxxfxxgx
y
g xgfx x xfx
′′′′
′=+=+
++
,
(4)
22
()
2() ()
2 () 4 () () () ()
2 () () 4 () () ()
gx
fxf x
g xfxfxgxgx
y
f xgx fxgxgx
′
′ +
′ ′+
′==
++
,
9,设 ()f x 在 (,)ll? 内可导,证明,如果 ()f x 是偶函数,则 ()f x′ 是奇函数 ; 如果 ()f x 是奇函数,则 ()f x′ 是偶函数,
证 如果 ()f x 是偶函数,则有 () ()f xfx? =,对等式两边对 x 求导,
有,() ()f xfx′′=,从而 () ()f xfx′′?=?,即 ()f x′ 是奇函数,
如果 ()f x 是奇函数,则有 () ()f xfx? =?,对等式两边对 x 求导,
有,() ()f xfx′′=?,从而 () ()f xfx′′?=,即 ()f x′ 是偶函数,
