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 csc x (2)(csc x)=-cscxcotx A2()(cot x),cos xy_(cos xy'sinx-(sin x)'cosssircsc2x (2)(cscx)’=( in x)=-cscx x sin x 2.求下列函数的导数 (3) y=2cscx+cotx (4) y=e arccos (7) y=xInxcosx 1+sin t (9)y=xa2(a>0) (2)y'=(n2x)+(2)+(x)(2x)+2hn2+1=-+2ln2+1 (3) y=(2cscx)+(cot x)=-2csc xcot x-cSc (4) y=(e)arccos x +e(arccos arccosr-e arccos (5)y=(x3)log2+x3(og2)=3x2log2+ x(3log2+,) xIn 2
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 ) csc x ′ = − x ; (2) (csc ) csc cot x ′ = − x x . 解 (1) 2 2 2 cos (cos ) sin (sin ) cos 1 (cot ) ( ) csc sin sin sin x xx x x x x x x x ′ − ′ ′ ′ = = =− =− . (2) 2 1 1 (csc ) ( ) (sin ) csc cot sin sin x x xx x x ′′ ′ = =− =− . 2. 求下列函数的导数: (1) 3 2 3 y x2 7 x = −+ ; (2) ln 2 2x yx x = + + ; (3) 2csc cot y xx = + ; (4) e arccos x y x = ; (5) 3 2 log x y x = ; (6) ln x y x = ; (7) 2 yx x x = ln cos ; (8) 2 2 1 1 x y x + = − ; (9) ( 0) a x y xa a = > ; (10) 1 sin 1 cos t s t + = + . 解 (1) 3 2 2 3 3 6 yx x (2 ) ( ) (7) 6 x x ′ ′ ′′ = − +=+ . (2) (2 ) 1 (ln 2 ) (2 ) ( ) 2 ln 2 1 2 ln 2 1 2 x xx x yx x x x ′ ′ ′ ′′ = + + = + += + + . (3) 2 y x x xx x ′ ′′ = + =− − (2csc ) (cot ) 2csc cot csc . (4) (e ) arccos e (arccos ) x x y xx ′′ ′ = + 2 2 1 1 e arccos e e (arccos ) 1 1 xxx x x x x =−= − − − . (5) 3 3 23 2 22 2 2 1 1 ( ) log (log ) 3 log (3log ) ln 2 ln 2 xx x x yx x x x x x ′′ ′ = + =+= +
6)y (7) y=(x)In xcos x+x"(In x) x+x"In x(cos x)" 2xln xcosx+xcos=.x (8)y=(1+￥21-x2)-(1+x2)(1-x)y4x (1-x2)2 (9)y=(x)a+x(a)=axa+xa Ina=xa(xIna+a) (10)P,-(1+sin/' (1+cost)-(1+sint)(1+cost)_1+cost+sint 3.以初速v竖直上抛的物体,其上升高度s与时间t的关系是s=v1-gr (1)该物体的速度v() (2)该物体到达最高点的时刻 (1)v(1)=s′=vo (2)物体到达最高点时,n()=0,即n0-gt=0,从而t=0 求曲线y=x(nx-1)上横坐标为x=e的点处的切线方程和法线方程 解该点为(e0),所求切线的斜率为y1-=(nx-1+1)-=hnx-=1,从而 切线方程为:y=x-e,法线方程为:y=-x+e 5.求下列函数的导数 (3) y=arctan(e); (4) y=(arcsin x) (6) y=cos(tanx) (7)y=2x 解(1) (-3x2)
2 (6) 2 2 (ln ) ln 1 ln x xx x y x x ′ − − ′ = = . (7) 22 2 y x x xx x xx x x ′′ ′ ′ = ++ ( ) ln cos (ln ) cos ln (cos ) 2 = +− 2 ln cos cos ln sin x x xx xx x x . (8) 2 2 22 22 22 (1 ) (1 ) (1 )(1 ) 4 (1 ) (1 ) x x xx x y x x + − −+ − ′ ′ ′ = = − − . (9) 1 1 ( ) ( ) ln ( ln ) a x a x a x ax a x y x a x a ax a x a a x a x a a − − ′′ ′ =+=+ = + . (10) 2 2 (1 sin ) (1 cos ) (1 sin )(1 cos ) 1 cos sin (1 cos ) (1 cos ) t t t t tt s t t + + −+ + + + ′ ′ ′ = = + + . 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 = − v t gt , 求: (1) 该物体的速度v t( ) ; (2) 该物体到达最高点的时刻. 解 (1) 0 v t s v gt ( ) == − ′ . (2) 物体到达最高点时, ( ) 0 v t = , 即 0 v gt − = 0 , 从而 0 v t g = . 4. 求曲线 yx x = − (ln 1) 上横坐标为 x = e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 e ee (ln 1 1) ln 1 x xx yx x = == ′ = −+ = = , 从而 切线方程为: e y x = − , 法线方程为: y x = − + e . 5. 求下列函数的导数: (1) 2 3 e x y − = ; (2) 2 y x = − cos(4 3 ) ; (3) arctan(e ) x y = ; (4) 2 y x = (arcsin ) ; (5) 2 tan ( 0) x ya a = > ; (6) 2 3 y x = cos (tan ) ; (7) 2 1 sin 2 x y = ; (8) x y x = . 解 (1) 2 2 32 3 e (3 ) 6e x x y xx − − ′ ′ = − =−
(3)y (4)y=2( )arcsin x) (5)y (6)y=2cos( tanx)lcos(tanx)I=-2cos(tanx) In 2 sir 1-Inx In x)= 6.求下列函数在指定点处的导数值 ( )f(o)=sin 3o (1-x) 解(1)r()=3c030+1-+2=3c083+1+,r(o)=4 (1-g2)2 (2)y 2x+ 5,y1=1=51-3e)
3 (2) 22 2 y x xx x ′ ′ =− − − = − sin(4 3 )(4 3 ) 6 sin(4 3 ). (3) 2 2 (e ) e 1e 1e x x x x y ′ ′ = = + + . (4) 2 2arcsin 2(arcsin )(arcsin ) 1 x y xx x ′ ′ = = − . (5) 2 2 2 tan 2 tan 2 2 2 2 2 tan ln (tan ) ln sec ( ) 2ln sec x x x y a a x a a x x a x xa ′′ ′ = = =⋅ . (6) 3 3 3 33 y x x x xx ′′ ′ = =− 2cos(tan )[cos(tan )] 2cos(tan )sin(tan )(tan ) 3 2 = −3sin(2 tan ) tan (tan ) x x x ′ 22 3 = −3tan sec sin(2 tan ) x x x . (7) 2 2 1 1 sin sin 2 1 11 2 ln 2(sin ) 2 2ln 2sin (sin ) x x y x x x ′′ ′ = = 2 2 1 1 sin sin 2 1 1 1 ln 2 2 2 2ln 2sin cos ( ) 2 sin x x x xx x x = =− ′ . (8) 1 1 ln ln 2 22 1 1 1 1 ln (e ) e ( ln ) ( ln ) x x x x x x x y xx x x x x xx − ′′ ′ = = =− += . 6. 求下列函数在指定点处的导数值: (1) 2 ( ) sin 3 , 0 1 f ϕ ϕϕ ϕ ϕ = + = − ; (2) 1 3 arcsin 2 , 4 y xx x = = ; (3) 5 3e 5(1 ), 1 x y xx − = − − =− . 解 (1) 22 2 22 22 12 1 ( ) 3cos3 3cos3 , (0) 4 (1 ) (1 ) f f ϕϕ ϕ ϕϕ ϕ ϕ ϕ −+ + ′ ′ =+ =+ = − − . (2) 3 2 2 4 1 2 16 arcsin 2 , (3 3 π ) 9 1 4 x y xy x x x = ′ ′ =− + = − − . (3) 5 5 1 15e 5, 5(1 3e ) x x y y − =− ′ ′ =− + = −
7.求下列函数的导数 (1) y=Insec x+ tan x) (2) y=In tan+arctan(tan - (3) y=sin"xcosnx (4) arcsin. 7)y= x arcsin+√4-x ( 8) y=sh-ch3x (9) y=In chx chx (10)y=a4+x“+a 解(1)y secx tanx+ sec 1 sec x secx+ tan x Isec cscw+ (3)y=nsin"-lxcox cosnx-nsin"xsin nx= nsin"-lxcos(n+1)x (4)y 1-(1+x)-(1-x) (1+x)2 cosx2sin2x-sinx22sin xcos x 2xcosx sin x-2sinx2cosx (5)y sin x (7) y=arcsin+x arcsin (8)y=ch-(--)ch3x+ 3sh-sh3x=--3ch-ch3x+ 3sh-sh3x (9)y= shx -thx(l h2x (10) y=a Ina(a)+ax-+a In a(x)
4 7. 求下列函数的导数: (1) ln(sec tan ) y xx = + ; (2) 1 ln tan arctan( tan ) 2 22 x x y = + ; (3) sin cos n y x nx = ; (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 y x sh ch3 x = ; (9) 2 1 ln ch 2ch y x x = + ; (10) xaa aax ya x a =++ . 解 (1) 2 sec tan sec sec sec tan xx x y x x x + ′ = = + . (2) 2 2 2 2 1 1 sec sec 2 24 2 1 csc 1 tan 1 tan 1 3cos 2 42 2 x x y x x x x ′ = + =+ + + . (3) 1 1 sin cos sin sin sin cos( 1) n nn y n xcox nx n x nx n x n x − − ′ = −= + . (4) 2 1 1 (1 ) (1 ) 1 1 1 (1 ) 2 (1 ) (1 ) 1 2 1 1 x x y x x xxx x x x − + −− ′ = =− −− + − + − + + . (5) 22 2 2 2 4 3 2 cos sin sin 2sin cos 2 cos sin 2sin cos sin sin x x x x xx xx x x x y x x − − ′ = = . (6) 22 2 2 2 12 1 (1 ) ( 1 ) 21 1 ( 1 ) x y x x x xx x ′ =− + =− ++ + + ++ . (7) 2 2 11 2 arcsin arcsin 22 2 2 4 1 4 x x x y x x x − ′ =+ + = − − . (8) 2 2 22 2 22 2 y xx xx ch ( )ch3 3sh sh3 ch ch3 3sh sh3 x x xx x x ′ = − + =− + . (9) 3 3 2 sh sh 1 2 th (1 ) th ch 2ch ch x x y xx x x x ′ =− = − = . (10) 1 ln ( ) ln ( ) x aa a x aa x a y a aa a x a ax − ′′ ′ = ++
In2 aatatax-ltaln axa-lar 8.设f(x)和g(x)都可导,求下列函数y的导数 dv (1)y=f(e (2) y=f(sin x)+f(cos x) (3) y=In f(vx)+arctan g(x) (4)y=Vf2(x)+√g(x) Af(1)y=f'(e )ee/()+f(e )e/(x)/'(x)=f'(e )e/()+x+f(e )e/(a)f'(x) 2)y=f(sin x)2sin xcos x-f(cos x)2sin cosx sin 2x[f(sin x)-f'(cos x)] g fx)2x1+g2(x2)2x/(x)1+g2(x2) 2(x)f(x)+8(x) (4)y 2√8(x)4/(x)/(xs(x)+8(x) /2(x)+√8(x) f2(x)+√g(x)√g(x) 9.设f(x)在(-,D)内可导,证明:如果f(x)是偶函数,则f(x)是奇函数;如 果∫(x)是奇函数,则f(x)是偶函数 证如果∫(x)是偶函数,则有f(-x)=f(x),对等式两边对x求导 有,-f(-x)=f(x),从而f(-x)=-f(x),即f(x)是奇函数 如果∫(x)是奇函数,则有f(-x)=-f(x),对等式两边对x求导 有,-f(-x)=-f(x),从而f(-x)=f(x),即f(x)是偶函数
5 2 11 ln ln xa a x a aa a x 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 f x y f = ; (2) 2 2 yf x f x = + (sin ) (cos ); (3) 2 y f x gx = + ln ( ) arctan ( ) ; (4) 2 y f x gx = + () () . 解 (1) () () () () (e )e e (e )e ( ) (e )e (e )e ( ) x x fx x fx x fx x x fx y f f fx f f fx + ′ =+ =+ ′ ′′ ′ . (2) 2 2 y f x x xf x x x ′′ ′ = − (sin )2sin cos (cos )2sin cos 2 2 = − sin 2 [ (sin ) (cos )] x f xf x ′ ′ . (3) 2 2 22 22 ( ) 1 ( )2 ( ) 2 ( ) ( )2 2 ( ) 1 () 1 () f x g x x f x xg x y f x x xf x g x gx ′ ′′′ ′ = += + + + . (4) 2 2 ( ) 2 () () 2 () 4 () () () () 2 () () 4 () () () g x fxf x g x f xf x gx g x y f x gx f x gx gx ′ ′ + ′ + ′ ′ = = + + . 9. 设 f ( ) x 在 ( ,) −l l 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 ( ) () −x fx = , 对等式两边对 x 求 导 , 有, ( ) () − −= f ′ ′ x fx , 从而 f ′ ′ ( ) () − =− x fx , 即 f ′( ) x 是奇函数. 如 果 f ( ) x 是奇函数 , 则 有 f ( ) () −x fx = − , 对等式两边对 x 求 导 , 有, ( ) () − − =− f ′ ′ x fx , 从而 f ′ ′ ( ) () − = x fx , 即 f ′( ) x 是偶函数