函數導數公式及證明

1、函數導數公式及證明函數類型原函數求導公式常量函數f(x) C, C為常量, If (x) 0哥函數f(x) xa(xa) axa 1(xa)(n) a(a 1)(a n 1)xa n(a 0,1,2,n 1)f(x) xm/ m(n)m!m n /、(x )7-x ,(n m)(m n)!指數函數f(x) ax(ax)axlna(ax)axlnn a, (0 a 1)f(x) ex(ex)ex(ex)ex對數函數f (x) lOga xZl、,1(lOga x) xln a八、(n)( 1)n1(n 1)! 小.、(logax)() v ；, (0 a 1)x ln af(x) ln x(ln

2、 x) xn 1/I 、(n)( 1) (n 1)！(ln x)j工-n1-x三角函數f (x) sinx，、(sin x) cosx(sinx)()sin(x )2f(x) cosx(cosx) sin x(cosx)(n) cos(x n-)f(x) tanx、21/ 、2(tanx) sec x 2 1 (tanx) cos xf(x) cotx,. 、21, 、2(cot x) csc x. 21 (cot x)sin x反三角函數f (x) arcsinx、1(arcsin x) ,2，1x2f(x) arccosx,、,1(arccos x)2V1 x2f (x) arctanx,

3、、,1(arctanx) 1 x2f(x) arccotx,、,1(arccotx) 21 x雙曲函數f (x) sinhx,. (sinh x) coshxf (x) coshx、(coshx) sinhxf (x) tanhx,.、,1(tanhx)2cosh xf (x) cothx,、,1(coth x). 2sinh x反雙曲函數f (x) arsinh x/. .、,_(arsinh x) TTVf (x) arcoshx，,、,1(ar cosh x) j 2vx2 1f (x) ar tanhx,、,1(ar tanhx)21 x復合函數導數公式復合函數求導公式f(x) g(x

4、)f(x) g(x) f (x) g (x)f(x)gg(x)f (x)gg(x) f (x)gg(x)f(x)gg (x)Cgf (x) Cgf (x)f(x)/ 、 cg (x) 0 g(x),f(x)f (x)gg(x) f(x)gg(x)2g(x)g (x)f g(x)f g(x) f (g(x)gg (x),1.證明哥函數f(x) xa的導數為f(x) (xa) axa 1 證:f(x)Vxm0 3VqlimUx)Vx 0 Vx根據二項式定理展開 (x Vx)nlim xn Cnxn 1VxVx 0C：xn2Vx2 C；1xVxn1 C；Vxn) xnVx消去C：xn xnC1xn

5、1Vx C2xn 2Vx2Cn 1xVxn 1CnVxnnx x -/n x x . x./n x xx-/n vxlim Vx 0Vx分式上下約去VxQmCnxn1C；xn 2Vx1 . C1 n 2xVxC：Vxn 1)n 2Vx)Vxn 2n 11.x (x Vx) x 因Vx 0 ,上式去掉零項C：xn nx lim (x Vx x)(x Vx)n 1 x(xVx 0JimJ( x Vx)n 1 x(x Vx)n 2 . xn 2(x Vx) xn 1n 2xg(2n 1gx xnng(2.證明指數函數f(x)ax的導數為(ax)axlna證:f (x)lim3工)Vx 0 Vxf(x

6、)x Vx x a a lim Vx 0 Vxx Vx.a (a 1) lim Vx 0 Vx令aVx1 m ,則有Vxloga(m 1),代入上式Vxm0x Vxa (a 1)Vxx一 a mlim Vx 0loga(m 1)xa m limVx 0 ln(m 1)ln aaxln a lim Vx 0 1 ln(m m1)axln alim t-Vx 01ln(m 1)m根據e的定義e lim(1x1貝U Jm( m 1)max ln alim 1Vx 01ln(m 1)maxln aln eax In a3.證明對數函數f (x) logax的導數為f (x) (loga x)1xln

7、a證:f(x)limfKMVx 0 VxB HmVx 0loga(x Vx)Vxloga xVx)xloga(1 limVx 0 Vx, x Vx logalim xVx 0 VxVx、 ln(1 ) lim-Vx 0 Vxlnaxx VxVXVxln(1 一) in(i 一產 lim Vxx lim x Vx 0 x in a Vx 0 x In a根據e的定義e lim(lx-)x xVx 2 一，則 Vxm*。7 Vx e In e 1xln a xlnaVx ln(1 Vx)Vx lim xVx 0 xln a4 .證明正弦函數f(x) sinx的導數為f(x) (sinx) cosx

8、證：jf (x Vx) f (x) sin(x Vx) sinxf (x) limlimVx 0VxVx 0Vx根據兩角和差公式 sin(x Vx) sin xcoSVx cosxsinVxlim sin(x Vx)Vx 0Vxsin xlimVx 0sinxcosVx cosxsinVx sinxVx因 Jmein xcosVx) sinx ,約去 sinxcosVx sinx,于是cosxsinVx lim Vx 0 Vx因 Vxm0sinVxVxcosxsinVx lim(cos xVx 0Vx5 .證明余弦函數f(x) cosx的導數為f(x) (cosx) sinx 證：f (x V

9、x) f (x) cos(x Vx) cosxf (x) lim lim Vx 0VxVx 0Vx根據兩角和差公式 cos(x Vx) cosxcosVx sinxsinVxcos(x Vx) cosxVxVixm0cosxcosVx sin xsinVx cosxVx因臚(cos xcosVx) cosx約去 cos x cosVxcosx,于Vxm0sin xsinVxVx因lim剪VxVx 0 VxVxm0(sin xsinVxVxsinx6.證明正切函數f(x)tanx的導數為f (x) (tan x)12- cos x證:f(x) lim 里乂Vx 0 Vxf(x)tan(x Vx)

10、 lim Vx 0VxtanxlimVx 0sin(x Vx)cos(x Vx)sin xcosxVxlim sin(x Vx)cosx sinxcos(x Vx)Vx 0 Vxcos(x Vx)cos x根據兩角和差公式sin(x Vx)sinxcosVxcosxsinVxcos(x Vx)cosxcosVxsin xsinVx代入上式(sinxcosVx cosxsinVx)cosx sinx(cosxcosVx sinxsinVx)Vxcos(x Vx)cosxcosxcosxsinVx ( sinxsinxsinVx)Vxcos(x Vx)cosxsinVx(cosxcosx sin

11、xsin x) limVx 0 Vxcos(x Vx)cos xsinVx因 VXm0sinVxVx1 ， lim cos(x Vx)Vx 0cosx,上式為vxm0sinVx 1Vx cos(x Vx)cos x12- cos x7.證明余切函數f (x) cotx的導數為f(x) (cotx)1sin2 x證:f(x)Vxm0f (x Vx) f (x)Vxcot(x Vx) cotx lim Vx 0Vxcos(x Vx) cosxsin(x Vx) sin x limVx 0Vxcos(x limVx 0Vx)sin x cosxsin(x Vx)Vxsin(x Vx)sin x根據兩

12、角和差公式sin(x Vx) sinxcosVx cosxsinVxcos(x Vx) cosxcosVx sin xsinVx代入上式limVx 0(cosxcosVx sinxsinVx)sin x cosx(sinxcosVx cosxsinVx)Vxsin(x Vx)sin xVxm0 Vxsin(x Vx)sin x.22sin xsinVx cos xsinVx22、sinVx(sin x cos x) lim Vx 0 Vxsin(x Vx)sin x因 sin2 xVim0sin(x Vx)sin X，代入上式Vxm0sinVxVx sin(x Vx)sin x12sin x8

13、 .證明復合函數f (x) g(x)的導數為f (x) g(x)f(x) g(x)f(x) g(x) lim f(x Vx) g(x Vx) f(x) g(x)Vx 0VxVxmoVxf(x Vx) f(x) g(x Vx) g(x)Vxf (x) g (x) 9 .證明復合函數 f(x)g(x)的導數為 f (x)g(x) f (x)g(x) f(x)g (x) 證：lim f(x Vx)g(x Vx) f(x)gVx 0Vxf (x Vx) f(x) f (x) g(x Vx) f (x) g(x Vx) g(x Vx) g(x) lim Vx 0Vx lim f(x Vx) f(x) g

14、(x Vx) f(x)g(x Vx) f(x)g(x Vx) f(x) g(x Vx) gVx 0VxVxm0f(x Vx) f(x) g(x Vx) f(x) g(x Vx) g(x)Vxlim f(x Vx) f(x)g(x Vx) f(x)g(x Vx) g(x) Vx 0 VxVxf (x)g(x) f(x)g (x)10.證明復合函數 9 的導數為f(x)gg(x)2 f(x)gg(x)g(x)g(x)g2(x)證：f(x Vx) f(x)f (x)g(x Vx) g(x) lim g(x) Vx 0 VxVxm0f(x Vx)g(x) f(x)g(x Vx)Vxg(x)g(x Vx

15、)f(x Vx) f(x) f(x) g(x) f(x) g(x Vx) g(x) g(x) lim V x 0Vxg (x)g(x Vx)f(x Vx) f(x) g(x) f (x)g(x) f (x) g(x Vx) g(x) f(x)g(x) lim V x 0Vxg(x)g(x Vx)f(x Vx) f (x) g(x) f(x) g(x Vx) g(x)lim V x 0Vxg(x)g(x Vx)limVx 0f(x Vx) f(x)Vxg(x)f(x)g(x Vx) g(x)vxg(x)g(x Vx) f (x)gg(x) f(x)gg(x)g2(x)11 .證明復合函數f g(

16、x)的導數為f g(x) f(g(x)gg(x) 證：f(g(x) lim f(g(x Vx) f(g(x) Vx 0Vx令 u g(x)，則有 Vu g(x Vx) g(x).f(u VU) f (u) lim -V x 0Vxlimf(uVu)f(u)VuV x 0 Vu Vxlimf(uVu)f(u)g(xVx) g(x)V x 0VuVxf (u)gg (x)f (x) f(x)f (g(x)gg (x)12 .證明復合函數ln f(x)的導數為ln f(x)令 u f(x),In f (x) Inu gjf (x)的13 .求復合函數xx的導數 解: 令u xxlnu xln x 等

17、式左邊求導為ln u等式右邊求導為xln x1x In x x(ln x) In x x- In x 1 x于是有In x 1, uu (In x 1)u則(xx)(In x 1)xx14.證明反三角函數arcsinx的導數為(arcsin x) J 、1 x2證：令 y arcsinx,則sin y x對上式兩邊求導，等式右邊x 1 等式左邊(根據受合函數求導公式工其導致為(sin y) (cosy)gy于是有(cosy)gy 111(cosy) 1 sin2 y再將 y arcsinx代入上式(arcsin x)12 /sin (arcsin x)15.證明反三角函數 arccosx 的導

18、數為(arccosx) ,1 x2令 y arccosx，則 cosy x 對上式兩邊求導，等式右邊x 1 等式右邊(根據復合函數求導公式)淇導數為cosy (siny)gy于是有 (sin y)gy 1，整理后如下:11y2(sin y) 1 cos y再將y arccosx代入上式(arccosx)1cos (arccos x)1_1 x2116.證明反二角函數arctanx的導數為(arctanx) 21 x令 y arctanx，則tany x對上式兩邊求導，等式右邊x 12、,等式右邊(根據復合函數求導公式)淇導數為tany (1 tan y)gy2于是有(1 tan y)gy1,整理后如下:1y 1 tan2 y再將y arctanx代入上式(arctanx)221 tan arctanx 1 x1,那么它的