函数导数公式及证明

1、函数导数公式及证明函数类型原函数求导公式常量函数f(x)C,C为常量,If(x)0哥函数f(x)xa(xa)'axa1(xa)(n)a(a1)(an1)xan(a0,1,2,n1)f(x)xm/m(n)m!mn/、(x)7-x,(nm)(mn)!指数函数f(x)ax(ax)'axlna(ax)axlnna,(0a1)f(x)ex(ex)'ex(ex)ex对数函数f(x)lOgaxZl、,1(lOgax)xlna八、(n)(1)n1(n1)!小.、(logax)()v；,(0a1)xlnaf(x)lnx(lnx)xn1/I、(n)(1)(n1)！(lnx)j工-n1-x三

2、角函数f(x)sinx，、'(sinx)cosx(sinx)()sin(x)2f(x)cosx'(cosx)sinx(cosx)(n)cos(xn-)f(x)tanx"、21/、2(tanx)secx21(tanx)cosxf(x)cotx,.、21,、2(cotx)cscx.21(cotx)sinx反三角函数f(x)arcsinx、1(arcsinx),2，1x2f(x)arccosx,、,1(arccosx)2V1x2f(x)arctanx,、,1(arctanx)1x2f(x)arccotx,、,1(arccotx)21x双曲函数f(x)sinhx,.'

3、(sinhx)coshxf(x)coshx、(coshx)sinhxf(x)tanhx,.、,1(tanhx)2coshxf(x)cothx,、,1(cothx).2sinhx反双曲函数f(x)arsinhx/.、,_(arsinhx)TTVf(x)arcoshx，,、,1(arcoshx)j2vx21f(x)artanhx,、,1(artanhx)21x复合函数导数公式复合函数求导公式f(x)g(x)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)0g(x),f(x)f(x)gg(x)f(x

4、)gg(x)2g(x)g(x)fg(x)fg(x)f(g(x)gg(x),1.证明哥函数f(x)xa的导数为f'(x)(xa)axa1证:f(x)Vxm03VqlimUx)Vx0Vx根据二项式定理展开(xVx)nlim ©xn Cnxn 1VxVx 0C：xn2Vx2C；1xVxn1C；Vxn)xnVx消去C：xnxnC1xn1VxC2xn2Vx2Cn1xVxn1CnVxnnxx-/nxx.x./nxxx-/nvxlimVx0Vx分式上下约去VxQmCnxn1C；xn2Vx1.C1n2xVxC：Vxn1)n 2Vx)Vxn 2n 11.x (x Vx) x 因Vx0,上式去掉

5、零项C：xnnxlim(xVxx)(xVx)n1x(xVx0JimJ(xVx)n1x(xVx)n2.xn2(xVx)xn1n 2xg(2n 1gx xnng(2.证明指数函数f(x)ax的导数为(ax)'axlna证:f (x)lim3工)Vx 0 Vxf(x)x Vx x a a lim Vx 0 VxxVx.a(a1)limVx0Vx令aVx1m,则有Vxloga(m1),代入上式Vxm0xVxa(a1)Vxx一amlimVx0loga(m1)1)xamlimVx0ln(m1)lnaaxlnalimVx01ln(mmaxlnalimt-Vx01ln(m1)m根据e的定义elim(1

6、x1贝UJm(m1)maxln aln eax In aaxlnalim1Vx01ln(m1)m3.证明对数函数f(x)logax的导数为f(x)(logax)1xlna证:f'(x)limfKMVx0VxBHmVx0loga(xVx)logaxVxVx)xloga(1 limVx 0 Vx,xVxlogalimxVx0VxVx、ln(1)lim-Vx0VxlnaxxVxVXVxln(1一)in(i一产limVxxlimxVx0xinaVx0xIna根据e的定义e lim(lx-)x xVx 2 一，则 Vxm*。7 Vx e In e 1xln a xlnaVxln(1Vx)Vxli

7、mxVx0xlna4 .证明正弦函数f(x)sinx的导数为f'(x)(sinx)'cosx证：jf(xVx)f(x)sin(xVx)sinxf(x)limlimVx0VxVx0Vx根据两角和差公式sin(xVx)sinxcoSVxcosxsinVxlim sin(x Vx)Vx 0Vxsin xlimVx 0sinxcosVx cosxsinVx sinxVx因JmeinxcosVx)sinx,约去sinxcosVxsinx,于是cosxsinVxlimVx0Vx因 Vxm0sinVxVxcosxsinVxlim(cosxVx0Vx5 .证明余弦函数f(x)cosx的导数为f

8、'(x)(cosx)'sinx证：f(xVx)f(x)cos(xVx)cosxf(x)limlimVx0VxVx0Vx根据两角和差公式cos(xVx)cosxcosVxsinxsinVxcos(xVx)cosxVxVixm0cosxcosVxsinxsinVxcosxVx因胪(cosxcosVx)cosx约去cosxcosVxcosx,于Vxm0sinxsinVxVx因lim剪VxVx0VxVxm0(sinxsinVxVxsinx6.证明正切函数f(x)tanx的导数为f(x)(tanx)12-cosx证:f'(x)lim里乂Vx0Vxf(x)tan(xVx)limVx

9、0VxtanxlimVx0sin(xVx)cos(xVx)sinxcosxVxlimsin(xVx)cosxsinxcos(xVx)Vx0Vxcos(xVx)cosx根据两角和差公式sin(x Vx)sinxcosVxcosxsinVxcos(x Vx)cosxcosVxsin xsinVx代入上式(sinxcosVxcosxsinVx)cosxsinx(cosxcosVxsinxsinVx)Vxcos(xVx)cosxcosxcosxsinVx ( sinxsinxsinVx)Vxcos(x Vx)cosxsinVx(cosxcosx sin xsin x) limVx 0 Vxcos(x

10、Vx)cos xsinVx因 VXm0sinVxVx1 ， lim cos(x Vx)Vx 0cosx,上式为vxm0sinVx 1Vx cos(x Vx)cos x12- cos x7.证明余切函数f (x) cotx的导数为f'(x) (cotx)1sin2 x证:f(x)Vxm0f (x Vx) f (x)Vxcot(x Vx) cotx lim Vx 0Vxcos(xVx)cosxsin(x Vx) sin x limVx 0Vxcos(x limVx 0Vx)sin x cosxsin(x Vx)Vxsin(x Vx)sin x根据两角和差公式sin(xVx)sinxcosV

11、xcosxsinVxcos(xVx)cosxcosVxsinxsinVx代入上式limVx 0(cosxcosVxsinxsinVx)sinxcosx(sinxcosVxcosxsinVx)Vxsin(xVx)sinxVxm0 Vxsin(x Vx)sin x.22sinxsinVxcosxsinVx22、sinVx(sinxcosx)limVx0Vxsin(xVx)sinx因sin2xVim0sin(xVx)sinX，代入上式Vxm0sinVxVxsin(xVx)sinx12sinx8 .证明复合函数f(x)g(x)的导数为f(x)g(x)f'(x)g'(x)f(x)g(x)

12、limf(xVx)g(xVx)f(x)g(x)Vx0VxVxmof(xVx)f(x)g(xVx)g(x)VxVxf(x)g(x)'9 .证明复合函数f(x)g(x)的导数为f(x)g(x)f(x)g(x)f(x)g(x)证：'limf(xVx)g(xVx)f(x)gVx0Vxf(xVx)f(x)f(x)g(xVx)f(x)g(xVx)g(xVx)g(x)limVx0Vxlimf(xVx)f(x)g(xVx)f(x)g(xVx)f(x)g(xVx)f(x)g(xVx)gVx0VxVxm0f(xVx)f(x)g(xVx)f(x)g(xVx)g(x)Vxlimf(xVx)f(x)g(

13、xVx)f(x)g(xVx)g(x)Vx0VxVxf(x)g(x)f(x)g(x)10.证明复合函数9的导数为f(x)gg(x)2f(x)gg(x)g(x)g(x)g2(x)证：f(xVx)f(x)f(x)g(xVx)g(x)limg(x)Vx0VxVxm0f(xVx)g(x)f(x)g(xVx)Vxg(x)g(xVx)f(xVx)f(x)f(x)g(x)f(x)g(xVx)g(x)g(x)limV x0Vxg(x)g(xVx)f(xVx)f(x)g(x)f(x)g(x)f(x)g(xVx)g(x)f(x)g(x)limV x0Vxg(x)g(xVx)f(xVx)f(x)g(x)f(x)g(x

14、Vx)g(x)limV x0Vxg(x)g(xVx)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 .证明复合函数fg(x)的导数为f'g(x)f'(g(x)gg'(x)证：f(g(x)limf(g(xVx)f(g(x)Vx0Vx令ug(x)，则有Vug(xVx)g(x).f(uVU)f(u)lim-V x0Vxlimf(uVu)f(u)VuV x0VuVxlimf(uVu)f(u)g(xVx)g(x)V x0VuVx''f(u)gg

15、(x)f (x) f(x)f(g(x)gg(x)12 .证明复合函数lnf(x)的导数为lnf(x)令uf(x),Inf(x)Inugjf(x)的13 .求复合函数xx的导数解:令uxxlnuxlnx等式左边求导为lnu等式右边求导为xln x1xInxx(lnx)Inxx-Inx1x于是有Inx1,uu(Inx1)u则(xx)(Inx1)xx14.证明反三角函数arcsinx的导数为(arcsinx)'J、1x2证：令yarcsinx,则sinyx对上式两边求导，等式右边x1''等式左边(根据受合函数求导公式工其导致为(siny)(cosy)gy于是有(cosy)gy

16、111(cosy)1sin2y再将yarcsinx代入上式(arcsin x)12/sin(arcsinx)15.证明反三角函数arccosx的导数为(arccosx),1x2令yarccosx，则cosyx对上式两边求导，等式右边x1等式右边(根据复合函数求导公式)淇导数为cosy(siny)gy于是有(siny)gy1，整理后如下:11y2(siny)1cosy再将yarccosx代入上式(arccosx)1cos (arccos x)1_1 x2116.证明反二角函数arctanx的导数为(arctanx) 21 x令 y arctanx，则tany x对上式两边求导，等式右边x 12、等式右边(根据复合函数求导公式)淇导数为tany (1 tan y)gy2于是有(1 tan y)gy1,整理后如下:1y 1 tan2 y再将y arctanx代入上式'(arctanx)221 tan arctanx 1 x1,17.证