函數(shù)導(dǎo)數(shù)公式證明

1、函數(shù)導(dǎo)數(shù)公式及證明函數(shù)種類(lèi)常量函數(shù)冪函數(shù)指數(shù)函數(shù)對(duì)數(shù)函數(shù)三角函數(shù)原函數(shù)f(x)C，C為常量f(x)xaf(x)xmf(x)axf(x)exf(x)logaxf(x)lnxf(x)sinxf(x)cosx求導(dǎo)公式f(x)0(xa)axa1(xa)(n)a(a1).(an1)xan(a0,1,2.,n1)(xm)(n)m!xmn,(nm)(mn)!(ax)axlna(ax)(n)axlnna,(0a1)(ex)ex(ex)(n)ex(logax)1xlna(logax)(n)(1)n1(n1)!,(0a1)xnlna(lnx)1x(lnx)(n)(1)n1(n1)!xn(sinx)cosx(sin

2、x)(n)sin(xn)2(cosx)sinx反三角函數(shù)雙曲函數(shù)反雙曲函數(shù)f(x)tanxf(x)cotxf(x)arcsinxf(x)arccosxf(x)arctanxf(x)arccotxf(x)sinhxf(x)coshxf(x)tanhxf(x)cothxf(x)arsinhxf(x)arcoshxf(x)artanhx(cosx)(n)cos(xn)2(tanx)sec2x1x1(tanx)2cos2(cotx)csc2x11(cotx)2sin2x(arcsinx)11x2(arccosx)11x2(arctanx)11x2(arccotx)11x2(sinhx)coshx(co

3、shx)sinhx(tanhx)1cosh2x(cothx)1xsinh2(arsinhx)1x21(arcoshx)1x21(artanhx)11x2復(fù)合函數(shù)導(dǎo)數(shù)公式復(fù)合函數(shù)f(x)g(x)f(x)gg(x)f(x)g(x)0fg(x)求導(dǎo)公式f(x)g(x)f(x)g(x)f(x)gg(x)f(x)gg(x)f(x)gg(x)Cgf(x)Cgf(x)f(x)gg(x)f(x)gg(x)f(x)g(x)g2(x)fg(x)f(g(x)gg(x),1證明冪函數(shù)f(x)xa的導(dǎo)數(shù)為f(x)(xa)axa1證：(x)f(xVx)f(x)(xVx)nxnflimVxlim0VxVx0Vx依照二項(xiàng)式定

4、理張開(kāi)(xVx)nlim(Cn0 xnCn1xn1VxCn2xn2Vx2.Cnn1xVxn1CnnVxn)xnVxVx0消去Cn0 xnxnlimCn1xn1VxCn2xn2Vx2.Cnn1xVxn1CnnVxnVxVx0分式上下約去Vxlim(Cn1xn1Cn2xn2Vx1.Cnn1xVxn2CnnVxn1)Vx0因Vx0，上式去掉零項(xiàng)Cn1xn1nxn1lim(xVxx)(xVx)n1x(xVx)n2.xn2(xVx)xn1Vx0Vxlim(xVx)n1x(xVx)n2.xn2(xVx)xn1Vx0 xn1xgxn2.xn2gxxn1ngxn12證明指數(shù)函數(shù)f(x)ax的導(dǎo)數(shù)為(ax)ax

5、lna證：f(x)limf(xVx)f(x)limaxVxaxVx0VxVx0Vxlimax(aVx1)VxVx0令aVx1m，則有Vxloga(m1)，代入上式limax(aVx1)limaxmVx1)Vx0Vx0loga(mlimaxmlimaxlnaaxlnaln(m1)01lim1Vx0Vxln(mVx01)mlnam1)ln(m1)x1依照e的定義elim(1，則lim(m1)me，于是xVx0 xlimaxlnaaxlnaxlna1lneaVx0ln(m1)m3證明對(duì)數(shù)函數(shù)fxlogax的導(dǎo)數(shù)為f(x)(logax)1()xlna證：f(x)limf(xVx)f(x)limloga

6、(xVx)logaxVx0VxVx0VxlogaxVxloga(1Vx)ln(1Vx)limVxxlimxlimxVx0Vx0VxVx0Vxlnaxln(1Vx)ln(1Vx)VxxlimVxxlim0 xVx0 xlnaVxxlna依照e的定義elim(11)x，則limln(1Vx)xVx0 xxxVxe，于是xln(1Vx)Vxlne1limxVx0 xlnaxlnaxlna4證明正弦函數(shù)f(x)sinx的導(dǎo)數(shù)為f(x)(sinx)cosx證：f(x)limf(xVx)f(x)limsin(xVx)sinxVx0VxVx0Vx依照兩角和差公式sin(xVx)sinxcosVxcosxsi

7、nVxlimsin(xVx)sinxlimsinxcosVxcosxsinVxsinxVx0VxVx0Vx因lim(sinxcosVx)sinx，約去sinxcosVxsinx，于是Vx0limcosxsinVxVx0Vx因limsinVx1，于是Vx0Vxlim(cosxsinVx)cosxVx0Vx5證明余弦函數(shù)f(x)cosx的導(dǎo)數(shù)為f(x)(cosx)sinx證：f(x)limf(xVx)f(x)limcos(xVx)cosxVx0VxVx0Vx依照兩角和差公式cos(xVx)cosxcosVxsinxsinVxlimcos(xVx)cosxlimcosxcosVxsinxsinVxc

8、osxVx0VxVx0Vx因lim(cosxcosVx)cosx，約去cosxcosVxcosx，于是Vx0limsinxsinVxVxVx0sinVx1，于是因limVxVx0lim(sinxsinVxsinxVx)Vx06證明正切函數(shù)f(x)tanx的導(dǎo)數(shù)為f(x)(tanx)1cos2x證：f(x)limf(xVx)f(x)limtan(xVx)tanxVx0VxVx0Vxsin(xVx)sinxlimcos(xVx)cosxlimsin(xVx)cosxsinxcos(xVx)Vx0VxVx0Vxcos(xVx)cosx依照兩角和差公式sin(xVx)sinxcosVxcosxsinV

9、x,cos(xVx)cosxcosVxsinxsinVx代入上式lim(sinxcosVxcosxsinVx)cosxsinx(cosxcosVxsinxsinVx)Vx0Vxcos(xVx)cosxlimcosxcosxsinVx(sinxsinxsinVx)sinVx(cosxcosxsinxsinx)Vxcos(xVx)cosxlim0Vxcos(xVx)cosxVx0Vx因cos2xsin2x1sinVxlimVx0Vxcos(xVx)cosx因limsinVx1，limcos(xVx)cosx，上式為Vx0VxVx0sinVx11lim2Vx0Vxcos(xVx)cosxcosx7證

10、明余切函數(shù)f(x)cotx的導(dǎo)數(shù)為f(x)(cotx)1sin2x證：f(x)limf(xVx)f(x)limcot(xVx)cotxVx0VxVx0Vxcos(xVx)cosxlimsin(xVx)sinxlimcos(xVx)sinxcosxsin(xVx)Vx0VxVx0Vxsin(xVx)sinx依照兩角和差公式sin(xVx)sinxcosVxcosxsinVx,cos(xVx)cosxcosVxsinxsinVx代入上式lim(cosxcosVxsinxsinVx)sinxcosx(sinxcosVxcosxsinVx)Vx0Vxsin(xVx)sinxlimsin2xsinVxc

11、os2xsinVxsinVx(sin2xcos2x)Vxsin(xVx)sinxlim0Vxsin(xVx)sinxVx0Vx因sin2xcos2x1,且limsinVx1，limsin(xVx)sinx，代入上式Vx0VxVx0limsinVx112Vx0Vxsin(xVx)sinxsinx8證明復(fù)合函數(shù)f(x)g(x)的導(dǎo)數(shù)為(x)g(x)f(x)g(x)f證:f(x)f(xVx)g(xVx)f(x)g(x)g(x)limVxVx0f(xVx)f(x)g(xVx)g(x)limVxVxVx0f(x)g(x)9證明復(fù)合函數(shù)f(x)g(x)的導(dǎo)數(shù)為f(x)g(x)(x)g(x)f(x)g(x)

12、f證:f(xVx)g(xVx)f(x)g(x)f(x)g(x)limVxVx0limf(xVx)f(x)f(x)g(xVx)f(x)g(xVx)g(xVx)g(x)Vx0Vxf(xVx)f(x)g(xVx)f(x)g(xVx)f(x)g(xVx)f(x)g(xVx)g(x)limVxVx0f(xVx)f(x)g(xVx)f(x)g(xVx)g(x)limVxVx0f(xVx)f(x)g(xVx)g(xVx)g(x)limVxf(x)VxVx0f(x)g(x)f(x)g(x)f(x)f(x)gg(x)f(x)gg(x)10證明復(fù)合函數(shù)的導(dǎo)數(shù)為f(x)g(x)g(x)g2(x)證:f(xVx)f(

13、x)g(xVx)g(x)f(x)g(x)limVxVx0f(xVx)g(x)f(x)g(xVx)limVx0Vxg(x)g(xVx)f(xVx)f(x)f(x)g(x)f(x)g(xVx)g(x)g(x)limVxg(x)g(xVx)Vx0f(xVx)f(x)g(x)f(x)g(x)f(x)g(xVx)g(x)f(x)g(x)limVxg(x)g(xVx)Vx0f(xVx)f(x)g(x)f(x)g(xVx)g(x)limVxg(x)g(xVx)Vx0f(xVx)f(x)g(x)f(x)g(xVx)g(x)limVxVxg(x)g(xVx)Vx0f(x)gg(x)f(x)gg(x)g2(x)1

14、1證明復(fù)合函數(shù)fg(x)的導(dǎo)數(shù)為fg(x)f(g(x)gg(x)證:f(g(x)limf(g(xVx)f(g(x)VxVx0令ug(x),則有Vug(xVx)g(x)limf(uVu)f(u)VxVx0limf(uVu)f(u)VuVuVxVx0limf(uVu)f(u)g(xVx)g(x)VuVxVx0f(u)gg(x)f(g(x)gg(x)12證明復(fù)合函數(shù)lnf(x)f(x)的導(dǎo)數(shù)為lnf(x)f(x)證:令uf(x),lnf(x)lnugu1guf(x)uf(x)13求復(fù)合函數(shù)xx的導(dǎo)數(shù)解:令uxxlnuxlnxu等式左邊求導(dǎo)為lnuuxlnxlnxx1lnx1等式右邊求導(dǎo)為xlnxx(

15、lnx)x于是有ulnx1,uu(lnx1)u則(xx)(lnx1)xx14.證明反三角函數(shù)arcsinx的導(dǎo)數(shù)為(arcsinx)11x2證:令yarcsinx,則sinyx對(duì)上式兩邊求導(dǎo),等式右邊x1等式左邊(依照復(fù)合函數(shù)求導(dǎo)公式),其導(dǎo)數(shù)為(siny)(cosy)gy于是有(cosy)gy1y11(cosy)1sin2y再將yarcsinx代入上式(arcsinx)111sin2(arcsinx)1x215.證明反三角函數(shù)arccosx的導(dǎo)數(shù)為(arccosx)11x2證:令yarccosx,則cosyx對(duì)上式兩邊求導(dǎo),等式右邊x1等式右邊(依照復(fù)合函數(shù)求導(dǎo)公式),其導(dǎo)數(shù)為cosy(siny)gy于是有(siny)gy1,整理后以下：y11(siny)1cos2y再將yarccosx代入上式(arccosx)111cos2(arccosx)1x216.證明反三角函數(shù)arctanx的導(dǎo)數(shù)為1x2(arctanx)1證:令yarctanx,則tanyx對(duì)上式兩邊求導(dǎo),等式右邊x12y)gy等式右邊(依照復(fù)合函數(shù)求導(dǎo)公式),其導(dǎo)數(shù)為tany(1tan于是有(1tan2y)gy1,整理后以下：y11tan2y再